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Guia De Estudo de C\303\241lculo \302\251 Maplesoft, a division of Waterloo Maple Inc., 2005 Cap\303\255tulo 17: An\303\241lise Local

Localizar o extremo global (extrema) \303\251 somente um passo na an\303\241lise de uma fun\303\247\303\243o (function). No presente cap\303\255tulo o interesse est\303\241 em determinar o comportamento local da fun\303\247\303\243o (function). Onde a curva est\303\241 crescendo (curve increasing) ou decrescendo? onde a curva \303\251 c\303\264ncava para cima (concave up) ou para baixo (down)? Em adi\303\247\303\243o, para definir estes termos, um importante conceito neste cap\303\255tulo \303\251 a conex\303\243o entre a primeira derivada (derivative) e uma fun\303\247\303\243o crescente/decrescente. Da mesma forma, existe uma conex\303\243o entre a segunda derivada (derivative) e concavidade. Estes fatos formam a base para identificar local de pontos de extremo ( extrema) e de inflex\303\243o( inflection). Os Testes de Primeira e Segunda Derivada s\303\243o introduzidos somente depois de se ter uma oportunidade para desenvolver alguma familiaridade com os novos conceitos. Estes teoremas prov\303\252em dois meios diferentes de olhar ao determinar se um ponto cr\303\255tico (critical point) \303\251 um local de m\303\241ximo ( local maximum) ou de m\303\255nima ( local minimum) -- ou nenhum deles. Os maplets An\303\241lise de Curva (CurveAnalysis) e Analisador de Fun\303\247\303\243o (FunctionAnalyzer) s\303\243o similares. Tanto um quanto o outro podem ser usados para apoiar as an\303\241lises que ser\303\243o necess\303\241rias neste cap\303\255tulo. A diferen\303\247a chave \303\251 que o maplet Analisador de Fun\303\247\303\243o(FunctionAnalyzer) pode produzir sinais (sign) indicadores (charts) para a primeira e segunda derivada. O maplet Gr\303\241fico de Derivada (DerivativePlot) \303\251 \303\272til para criar um gr\303\241fico (graph) contendo quase todas as combina\303\247\303\265es da fun\303\247\303\243o e suas derivadas.

Os seguintes termos, neste cap\303\255tulo, t\303\252m conex\303\243o com o Dicion\303\241rio de Matem\303\241tica Maple ( Maple Math Dictionary). absolutevalue asymptote bolzanostheorem interval concavedown concaveup concavity constant continuous criticalpoint curve decreasing derivative difference differentiable discontinuity domain evaluate explicit extremum factor floatingpoint function minimum graph hypothesis increasing independent variable inequality inflectionpoint interval jumpdiscontinuity linear maximum minimum multiple negative optimization period polynomial positive range realline real satisfy set sign singularpoint singularity slope solution solve tangent trigonometricfunction value value y-axis work zero

restart; with( plots ): with(Student[Calculus1]):

Inserir o cursor em qualquer lugar no c\303\263digo Maple a seguir e pressione a chave Enter InInterval := proc(x,S::list(numeric)) if has(x,I) then return false end if; return evalb( evalf(x)>=op(1,S) and evalf(x)<=op(2,S) ) end proc: TestIncrDecr := proc( df, pts ) local c, m, fmt1, fmt2; Digits := 4; fmt1 := "f'(%a)=%a; the function is %a on the subinterval containing %a"; fmt2 := "f'(%a)=%a; the function has a stationary point at x = %a"; for c in evalf(pts) do m := eval( df, x=c ); if m>0 then print( printf(fmt1, c, m, `increasing`, c) ); elif m<0 then print( printf(fmt1, c, m, `decreasing`, c) ); else print( printf(fmt2, c) ); end if; end do: end proc: TestConcavity := proc( d2f, pts ) local a, c, fmt1, fmt2; Digits := 4; fmt1 := "f''(%a)=%a; the function is concave %a on the subinterval containing %a"; fmt2 := "f''(%a)=%a; the function has a possible inflection point at x = %a"; for c in evalf(pts) do a := eval( d2f, x=c ); if a>0 then print( printf(fmt1, c, a, `up`, c) ); elif a<0 then print( printf(fmt1, c, a, `down`, c) ); else print( printf(fmt2, c, a, c) ); end if end do: end proc:

Como um gr\303\241fico \303\251 tra\303\247ado da esquerda para direita, sua tend\303\252ncia para subir ou cair \303\251 descrito pelos termos crescente e decrescente, respectivamente. Estes termos s\303\243o mais exatos na Defini\303\247\303\243o 17.1. Defini\303\247\303\243o 17.1: Fun\303\247\303\265es Crescentes e Decrescentes Seja f uma fun\303\247\303\243o definida no intervalo (interval) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . f \303\251 crescente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= se os valores da fun\303\247\303\243o se tornam maiores \303\240 medida que vari\303\241vel independente (independent variable) se torna maior , i.e., para cada par de n\303\272meros LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIxRidGL0YvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0Yv e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIyRidGL0YvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0Yv em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIxRidGL0YvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0Yv < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIyRidGL0YvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0Yv , \303\251 verdade que 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 > 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

Uma fun\303\247\303\243o que \303\251 constante (constant) em qualquer subintervalo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= n\303\243o \303\251 nem crescente nem decrescente (decreasing). Note que as desigualdades (inequalities) s\303\243o estritas. Uma fun\303\247\303\243o \303\251 chamada de n\303\243o decrescente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIxRidGL0YvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0Yv < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIyRidGL0YvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0Yv implica 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. Uma fun\303\247\303\243o \303\251 chamada de n\303\243o cresncente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIxRidGL0YvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0Yv < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIyRidGL0YvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0Yv implica 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 > 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.

A defini\303\247\303\243o de uma fun\303\247\303\243o crescente ou decrescente \303\251 bastante intuitiva. Entretanto, \303\251 mais f\303\241cil aplicar quando um gr\303\241fico da fun\303\247\303\243o est\303\241 dispon\303\255vel. como 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 \303\251 a inclina\303\247\303\243o (slope) da linha tangente ao gr\303\241fico de 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 co ponto 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, a primeira derivada deve ser \303\272til para determinar se uma fun\303\247\303\243o est\303\241 crescendo ou decrescendo em um intervalo (interval) contendo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. A conex\303\243o formal entre crescente/decrescente e a primeira derivada est\303\241 contida no Teorema 17.1. Teorema 17.1: Teste de Crescente/Decrescente Increasing/Decreasing Test Seja f uma fun\303\247\303\243o que \303\251 cont\303\255nua no intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e diferenci\303\241vel em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (exceto possivelmente nos limites). Ent\303\243o: se 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 > 0 para qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (exceto possivelmente nos limites), ent\303\243o f \303\251 crescente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Jy1JI21pR0YkNiVRJGZ+J0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GLDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZELUYjNiYtRiw2LVEiKEYnRi8vRjNGTkY1L0Y4Rk5GOUY7Rj1GPy9GQlEsMC4xNjY2NjY3ZW1GJy9GRUZmbi1GIzYkLUZJNiVRInhGJ0ZMRk9GLy1GLDYtUSIpRidGL0ZZRjVGWkY5RjtGPUY/RmVuL0ZFUSwwLjExMTExMTFlbUYnRi9GK0YvRitGLw== < 0 para qualquerl LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (exceto possivelmente nos limites), ent\303\243o f \303\251 decrescente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

O sinal da derivada nos limites (do intervalo) (endpoints) n\303\243o \303\251 importante. De fato, n\303\243o \303\251 nenhum problema se a derivada n\303\243o exista em um ou nos dois limites (endpoints).

Encontre os intervalos nos quais a fun\303\247\303\243o f1 := x^3 - 3*x^2 - x: f(x) = f1; \303\251 crescente e decrescente. Por que todos os polin\303\264mios (polynomials) s\303\243o cont\303\255nuos (continuous) em toda reta real, esta fun\303\247\303\243o -- e todas suas derivadas -- s\303\243o automaticamente cont\303\255nuas para qualquer valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . Relembre que no Exemplo 16.5 do Cap\303\255tulo 16 encontramos dois pontos estacion\303\241rios para esta fun\303\247\303\243o. A derivada 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 \303\251 df1 := diff( f1, x ): `f'`(x) = df1; e as solu\303\247\303\265es (solutions) da equa\303\247\303\243o 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 s\303\243o stat1 := solve( df1 = 0, x ): stat1; Por que a derivada \303\251 cont\303\255nua, o Teorema do Valor M\303\251dio ( Intermediate Value Theorem ) nos permite concluir que a derivada ter\303\241 um sinal em cada um dos tr\303\252s intervalos seguintes : ( LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2JS1GLDYtUSomdW1pbnVzMDtGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZELUkjbWlHRiQ2JVEoJiM4NzM0O0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YvRitGLw== , 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 ), ( 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Jy1JI21uR0YkNiRRIjFGJ0YvLUYsNi1RIitGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRicvRkVGUC1JJm1mcmFjR0YkNigtRiM2JC1GSTYkUSIyRidGL0YvLUYjNiZGKy1JJm1zcXJ0R0YkNiMtRkk2JFEiM0YnRi9GK0YvLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zhby8lKWJldmVsbGVkR0Y0RitGL0YrRi8= ), and ( 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM4NzM0O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ) Para determinar em quais destes intervalos a fun\303\247\303\243o \303\251 crescente e em quais \303\251 decrescente, simplesmente avalie (evaluate) a derivada em um ponto conveniente em cada subintervalo. Por exemplo, o comando TestIncrDecr, definido na se\303\247\303\243o de inicializa\303\247\303\243o, avalia a primeira derivada em cada ponto e usa o sinal do resultado para informar se a fun\303\247\303\243o \303\251 crescente, decrescente ou estacion\303\241ria no subintervalo contendo o ponto. O uso deste comando d\303\241 testpts1 := [ -1, 0, 3 ]: TestIncrDecr( df1, testpts1 ); Logo, a fun\303\247\303\243o 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 \303\251 crescente em (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2JS1GLDYtUSomdW1pbnVzMDtGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZELUkjbWlHRiQ2JVEoJiM4NzM0O0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YvRitGLw== , 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 ) e (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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM4NzM0O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ) e decrescente em (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Jy1JI21uR0YkNiRRIjFGJ0YvLUYsNi1RKCZtaW51cztGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRicvRkVGUC1JJm1mcmFjR0YkNigtRiM2JC1GSTYkUSIyRidGL0YvLUYjNiZGKy1JJm1zcXJ0R0YkNiMtRkk2JFEiM0YnRi9GK0YvLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zhby8lKWJldmVsbGVkR0Y0RitGL0YrRi8= , 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 ).

Maple's solve command can solve some inequalities. In this case, it is able to provide a direct answer to our question. The function is increasing on solve( df1 > 0, x ); and decreasing on solve( df1 < 0, x );

Figura 17.1 prov\303\252 um gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRImZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1JKG1mZW5jZWRHRiQ2JC1GIzYkLUZJNiVRInhGJ0ZMRk9GL0YvRi9GK0Yv e sua derivadas, Com os intervalos crescentes mostrados em vermelho e o intervalo decrescente mostrado em azul. p1 := plot( f1, x=-2 ..1-2/sqrt(3), color=red, legend=["y = f(x): increasing"] ): p2 := plot( f1, x=1-2/sqrt(3)..1+2/sqrt(3), color=blue, legend=["y = f(x): decreasing"] ): p3 := plot( f1, x=1+2/sqrt(3)..4, color=red, legend=["y = f(x): increasing"] ): p4 := plot( df1, x=-2 ..4, color=green, legend=["y = f '(x)"] ): display( [p1,p2,p3,p4] ); Figura 17.1 Um gr\303\241fico de 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 e suas derivadas, com intervalos de crescimento em vermelho e de decrescimento, em azul Observe como o gr\303\241fico da derivada muda de sinal precisamente onde a fun\303\247\303\243o muda de crescente para decrescente e depois de decrescente para crescente.

Existe um n\303\272mero de maplets que podem ser usados para obter informa\303\247\303\243o sobre onde uma fun\303\247\303\243o \303\251 crescente e decrescente.

O maplet An\303\241lise de Curva (CurveAnalysis maplet) (equivalente a inserir o Tutor de An\303\241lise de Curva (Curve Analysis Tutor), acess\303\255vel do Ferramentas/Tutores/C\303\241lculo - menu de uma vari\303\241vel (Tools/Tutors/Calculus - Single Variable menu)) pode explicitamente identificar intervalos nos quais a fun\303\247\303\243o \303\251 crescente ou decrescente. Figura 17.2 mostra o maplet aplicado \303\240 fun\303\247\303\243o de exemplo 17.1. 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Figura 17.2 Exemplo 17.1: Tutor de An\303\241lise de Curva (Curve Analysis Tutor) aplicado a 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 Para usar este maplet para determinar intervalos crescentes da fun\303\247\303\243o To use this maplet to determine the increasing intervals for a function Entre com a fun\303\247\303\243o (como uma fun\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ) e o intervalo (\303\251 melhor evitar o uso de infinitos mas algumas vezes trabalha com o m\303\255nimo de obje\303\247\303\265es) Escolha o bot\303\243o Decrescente (Increasing) pressione o bot\303\243o Exibir (Display) O intervalo(s) no qual a fun\303\247\303\243o \303\251 crescente ser\303\241 mostrado na caixa grande. Note que todos os intervalos aparecem ser intervalos fechados. Obtendo o intervalo(s) onde a fun\303\247\303\243o \303\251 decrescente deve aparecer. Outras caracter\303\255sticas deste maplet ser\303\243o discutidas mais tarde neste cap\303\255tulo.

O maplet Analizador de Fun\303\247\303\243o (FunctionAnalyzer maplet) \303\251 similar. Tem um flexibilidade um pouco maior mas requer um pouco mais do usu\303\241rio. Aplicado \303\240 fun\303\247\303\243o 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produz os resutados mostrados na Figura 17.3. 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qySGty<T]isyGjkwy:fkYVutv@GBGJx]PuyRREqvlshus:qqmLPeUOIeKb=TUaQCTo>yLFaLtTVeYvDMWjHpn=oy=J[qlUTsKxY:LrgDNf=t<xopYJ[=pkPxxYw^yi;vrvfx`QqihwWX]xYjTZAqynvif?wYaxRVty?xhvakepwSjMhsaehwT\\KWt_T@csqWBtgnlmlKdvY=P;ljj<Ns=uKARbmJjYpTTqoxOBxWJPuWYucqxpqwkEM^Xu@ILOlT?myQTsVYLqdMAQuOqySIty<wkp>;@Gx^qyRIty;BbGx`<:FxNWiA?uVWks`iVPlIG]s>nOy[gimY:ryPVcxif?wyAZny\\Myijq?wiDaGEestmSGub@WilICK_FS?xF?FO[CBSvrsNehqYeRGUnylxpMMCPVVMwaxtxAqOyY:hWGxmx;xiD@neOayfYZvQg]y`ONn_nrhyvev^?hfMpiyAZD>g>Gq^vmgy\\>^f<@h>GeK?r[Aht^kZ?l^oaVVql^`Y@xF@g[Gm=P\\tXki@b:WZK>hvxmNHfHoc;gsnwy:Vutv@G:BTwYp:;BrrMohI?HsaIFIyneWdohWyfJ]HcQuZmB`eWxwSRSfsscyIIxYdBKwL=YLEsfqTHsUC=ysiGJ=yRiSe;F@?vjuyO?tKAsiATj?chyWuh;Zx=:Fx]l[MxsamWqP?tygqy]TRt]m]esS=piTm;tTu<S@lokHQjljA\\SHMPGqunXJ>qtP@pTQUcMUN=p<MjrTNGIr<Pn;<jAlWtEjTyUUMqqdY@:kYnuyfAOb;bkYFTqy;IS`qDLeWoWbbiipoN[QNVYUA]MmDMlyPi]oRQo]=N:<juamPAQdxQUMat=vy:f\\AkAyY:@n@K;JketGpPhaPfxu\\<OyawbyY:\\xRTD;avidYVUy;jyOkcH]rWx?yd@XaRPvHxal>ufibVgwx?h:nyOx[pqsIlZpA;:>O::>ZSgrIaynipyeiwD]styxv_xowQXoYEj@;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::jMyQjmoy::::::::G?:::::::::::::::::::::::hmyO:::::::::i;::I?::::::::G?:::::::::::::::::::::::::::::]f_CYdwRSYGBYfheyQqySIty<D<j:krwifsyLYFHGnY;fECthACLayh?eDII@WH:mrk=yMOwAqhHfZnyUQ:OvbqySI<:j:@srILyQ^xaxYPYuS?atkoi_yvV=WXgdpEFa_iU[FuOVruRfgYTmIjmcbiwc?w>sc\\qhPgX=cWXexekIJIbxADwARZgHcwYpexy;BtQSYyI:=v=wIG]TOcx:qQfxY:<tPYYyMjY\\ybCeyEngtxyQ;xKx]PKAr=TYaAuyGbUwwvA[yGx]Xya;:PN:xkgem]qy<xpY=yJxY:Tfgim]oy<tqkI>yI:AYKcB:yEUQyAcY?wwy;BjIvy;b@ycgeu]djiyQjInygkN>d;fym<:tiPyabY_w;BJIbwW@=<:::::TF:vY::;:bg::@::::::FB:::::bk:jy;:>:Zp<:R::::::jZ:::::Z@=:yA:J::TE:Z;::::::=<:::::TF:vY::;:bg::@::::::FB:::::bk:jy;:>:Zp<:R::::::jZ:::::Z@=:yA:J::TE:Z;::::::=<:::::TF:vY::;:bg::@::::::FB:::::bk:jy;:>:Zp<:B;ib`WheSeUcbtiGkcCUSgI[Hy?FF=ts=G;OFD?gd?sS`G::5:\"\{\}-%%ROOTG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YGF'-%-BOUNDS_WIDTHG6#$"%+gF*-%.BOUNDS_HEIGHTG6#$"%SVF*-%)CHILDRENG6" Figura 17.3 O maplet FunctionAnalyzer aplicado a 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 Para usar o maplet Analizador de Fun\303\247\303\243o (FunctionAnalyzer) para identificar os intervalos nos quais a fun\303\247\303\243o \303\251 crescente entre co a fun\303\247\303\243o na regi\303\243o denominada y = no painel Fun\303\247\303\243o (Function) as primeira e segunda derivadas ser\303\243o computadas automaticamente No painel Resolva uma Equa\303\247\303\243o ou Inequa\303\247\303\243o (Solve an Equation or Inequality) l, entre com Dy > 0 no campo demominado Entre a equa\303\247\303\243o para ser resolvida (Enter equation to be solved) (use y, Dy, D2y) pressione o bot\303\243o Tentativa para resolver equa\303\247\303\243o (Attempt to solve equation) Os resultados do Maple ser\303\243o exibidos na regi\303\243o imediatamente acima do bot\303\243o que acabou de ser clicado. A \303\272nica caracter\303\255stica do maplet Analizador de Fun\303\247\303\243o (FunctionAnalyzer) \303\251 a habilidade de criar um " sinal indicador" ("sign chart") para a primeira derivada de uma fun\303\247\303\243o. Um " sinal indicador" pode ser um m\303\251todo particularmente efetivo para exibir intervalos onde uma fun\303\247\303\243o \303\251 crescente ou decrescente. Para ativar esta caracter\303\255stica clique na caixa Mostre o sinal indicador (Show sign chart box) no painel Primeira Derivada (First Derivative) nos campos a direita do nome Eixo Horizontal( Horizontal Axis), entre com os limites do intervalo e o nome da vari\303\241vel independente pressione o bot\303\243o Atualizar Gr\303\241fico (Update plot) A janela gr\303\241fica mostrar\303\241 um conjunto (set) vazio de eixos exceto por uma reta horizontal pr\303\263xima do fundo. A fun\303\247\303\243o \303\251 crescente onde esta reta for vermelha e \303\251 decrescente onde eata reta for azul. Este \303\251 o sinal indicador para a primeira derivada. Para incluir a fun\303\247\303\243o e/ou a primeira derivada no gr\303\241fico, clique os bot\303\265e apropriados Incluir no Gr\303\241fico (Include in plot) nos pain\303\251is Fun\303\247\303\243o (Function) e Primeira Derivada (First Derivative), depois pressione o bot\303\243o Atualizar Gr\303\241fico (Update plot).

O maplet Gr\303\241fico de Derivada ( DerivativePlot maplet) (equivalente a inserir Tutor de Derivada (Derivative Tutor) dispon\303\255vel do Ferramentas/Tutores/C\303\241culo (Tools/Tutors/Calculus) - menu Uma Vari\303\241vel (Single Variable) como Derivadas (Derivatives)...) tem um prop\303\263sito muito simples - para uma dada fun\303\247\303\243o, criar um gr\303\241fico mostrando quase toda combina\303\247\303\243o da fun\303\247\303\243o e suas derivadas. Por exemplo, para mostrar um gr\303\241fico de uma fun\303\247\303\243o e suas derivadas Entre com a Fun\303\247\303\243o (Function) (n\303\243o \303\251 necess\303\241rio entrar com o ranque ( range) nos campos a = e b = ) pressione o bot\303\243o Plot A Janela Gr\303\241fica (Plot Window) ser\303\241 atualizada para conter um gr\303\241fico da fun\303\247\303\243o e sua primeira derivada no intervalo especificado. A Aplica\303\247\303\243o da fun\303\247\303\243o 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 do Exemplo 17.1, resulta na Figura 17.4. 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Figura 17.4 Aplica\303\247\303\243o do Tutor de Derivada (Derivative Tutor) para 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

Encontre os intervalos nos quais a fun\303\247\303\243o f2 := (x^3 - 3*x^2 - x)/(x^2-4): f(x) = f2; \303\251 crescente e decrescente. oEsta fun\303\247\303\243o \303\251 cont\303\255nua e diferenci\303\241vel (differentiable) para todo n\303\272mero real (real numbers) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= exceto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFEiMkYnRi9GL0YrRi8= e 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. As singularidades (singularities) em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFEiMkYnRi9GL0YrRi8= e 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 precisa ser lembrado porque estes s\303\243o os pontos onde a derivada pode saltar de positiva ( positive) para negativa, ou vice versa sing2 := -2, 2: sing2; A derivada \303\251 df2 := simplify(diff( f2, x )): `f'`(x) = df2; Dos quatros zeros (zeroes) do numerador, existem dois que s\303\243o solu\303\247\303\265es reais. Estes s\303\243o pontos estacion\303\241rios. q1 := solve( numer(df2) = 0, x ): stat2 := op(remove( has, allvalues([q1]), I )): evalf[4](stat2); Pela continuidade da derivada e o Teorema do Ponto M\303\251dio ( Intermediate Value Theorem), a derivada pode mudar de sinal em um ponto onde a derivada \303\251 zero (um ponto estacion\303\241rio) ou onde a fun\303\247\303\243o e/ou derivada n\303\243o \303\251 definida (um ponto singular). Portanto, a reta real (real line) \303\251 dividida nos seguintes cinco intervalos nos quais a fun\303\247\303\243o \303\251 crescente ou decrescente: 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 Para determinar em quais destes intervalos a fun\303\247\303\243o \303\251 crescente e em quais ela \303\251 decrescente, simplesmente avalie a derivada em um ponto conveniente em cada subintervalo. testpts2 := [ -5, -3, -1, 1, 3 ]: TestIncrDecr( df2, testpts2 ); Portanto, a fun\303\247\303\243o 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 \303\251 crescente em 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 e decrescente emLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21uR0YkNiRRLiZ1bWludXMwOzQuMDhGJ0YvLUYsNi1RIixGJ0YvRjIvRjZRJXRydWVGJ0Y3RjlGO0Y9Rj9GQS9GRVEsMC4zMzMzMzMzZW1GJy1GTDYkUSsmdW1pbnVzMDsyRidGL0YvRi9GK0Yv e 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Por que 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 n\303\243o pertence ao dom\303\255nio (domain) desta fun\303\247\303\243o, seria incorreto dizer que esta fun\303\247\303\243o \303\251 crescente em ( -0.16, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM4NzM0O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ).

Usando o comando Maple solve, foi encontrado que a fun\303\247\303\243o \303\251 crescente em solve( df2 > 0., x ); e decrescente em solve( df2 < 0., x ); Note o uso do zero com ponto flutuante, 0., para for\303\247ar o Maple informar os resultados com ponto flutuante( floating-point) Se o inteiro zero, 0, for usado, a sa\303\255da (reposta) do Maple \303\251 bastante impositiva -- mas exata! (Tente!)

Figura 17.5, prov\303\252 um gr\303\241fico desta fun\303\247\303\243o, com intervalos crescentes mostrados em vermelho e o o intervalo decrescente mostrado em azul. p1 := plot( f2, x=-16 ..-4.08, color=red): p2 := plot( f2, x=-4.08..-2, color=blue): p3 := plot( f2, x=-2 ..-0.16, color=blue): p4 := plot( f2, x=-0.16.. 2, color=red): p5 := plot( f2, x= 2 ..16, color=red): p6 := plot( df2, x=-16 ..16, color=green, discont=true): display([p||(1..6)], view=[-16..16,-20..20]); plot([0$6],x=0..1, .1..0.2, scaling=constrained, legend=["y = f(x): increasing", "y = f(x): decreasing", "y = f(x): decreasing", "y = f(x): increasing", "y = f(x): increasing", "y = f '(x)"], color=[red,blue,blue,red,red,green]); Figura 17.5 Exemplo 17.2: Gr\303\241fico de 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 com intervalos de crescimento em vermelho e decrescimento, em azul Observe como a fun\303\247\303\243o muda de crescente para decrescente nos dois pontos estacion\303\241rios. Note tamb\303\251m que em 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 a fun\303\247\303\243o \303\251 decrescente em qualquer dos dois lados mas n\303\243o decrescente em qualquer intervalo contendo pontos de ambos os lados da singularidade. O mesmo comportamento pode ser visto em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFEiMkYnRi9GL0YrRi8= exceto que a fun\303\247\303\243o \303\251 crescente em ambos os lados desta singularidade.

Encontre os intervalos onde a fun\303\247\303\243o f3 := x^5-7*x^4+10*x^3+18*x^2-27*x-27: f(x) = f3; \303\251 crescente e decrescente. Figura 17.6 mostra a Janela Gr\303\241fica do maplet Analizador de Fun\303\247\303\243o (FunctionAnalyzer ). A curva vermelha \303\251 a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRImZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1JKG1mZW5jZWRHRiQ2JC1GIzYkLUZJNiVRInhGJ0ZMRk9GL0YvRi9GK0Yv, a verde, sua primeira derivada. O sinal indicativo \303\251 vermelho onde a fun\303\247\303\243o \303\251 crescente, e azul onde ela \303\251 decrescente. 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Figura 17.6 Exemplo 17.3: Janela Gr\303\241fica do maplet Analisador de Fun\303\247\303\243o (FunctionAnalyzer), usando magenta para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e verde para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmcHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ==. O sinal indicador para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmcHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== usa vermelho para positivo e azul para negativo. Porque todos os polin\303\264mios s\303\243o cont\303\255nuos em toda reta real, esta fun\303\247\303\243o -- e todas as suas derivadas -- s\303\243o automaticamente cont\303\255nuas para todos os valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. A derivada \303\251 o polin\303\264mio de quarto grau df3 := diff( f3, x ): `f'`(x) = df3; do qual as (quatro) ra\303\255zes s\303\243o stat3 := solve( df3 = 0, x ): stat3; Por que a derivada \303\251 cont\303\255nua, o Teorema do Ponto M\303\251dio ( do ponto m\303\251dio?)(Intermediate Value Theorem) permite concluir que a derivada ter\303\241 um sinal em cada um dos quatro intervalos seguintes: 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 Para determinar em quais destes intervalos a fun\303\247\303\243o \303\251 crescente e em quais \303\251 decrescente, simplesmente avalie a derivada em um ponto conveniente em cada subintervalo. testpts3 := [ -2, 0, 2, 4 ]: TestIncrDecr( df3, testpts3 ); Portanto, a fun\303\247\303\243o 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 \303\251 crescente em 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 e decrescente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRKyZ1bWludXMwOzFGJ0YvLUYsNi1RIixGJ0YvRjIvRjZRJXRydWVGJ0Y3RjlGO0Y9Rj9GQS9GRVEsMC4zMzMzMzMzZW1GJy1JJm1mcmFjR0YkNigtRkw2JFEiM0YnRi8tRkw2JFEiNUYnRi8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl5vLyUpYmV2ZWxsZWRHRjRGK0YvRi8tRiw2LVEiLkYnRi9GMkY1RjdGOUY7Rj1GP0ZBRkRGLw== Porque a fun\303\247\303\243o \303\251 definida em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFEiM0YnRi9GL0YrRi8= , seria correto dizer que a fun\303\247\303\243o \303\251 crescente emLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSShtZmVuY2VkR0YkNiQtRiM2J0YrLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjNGJ0YvLUZPNiRRIjVGJ0YvLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZaLyUpYmV2ZWxsZWRHRjQtRiw2LVEiLEYnRi9GMi9GNlEldHJ1ZUYnRjdGOUY7Rj1GP0ZBL0ZFUSwwLjMzMzMzMzNlbUYnLUkjbWlHRiQ2JVEoJmluZmluO0YnLyUnaXRhbGljR0Zdby9GMFEnaXRhbGljRidGL0YvRitGLw==. Este ponto ser\303\241 retomado (addressed again) depois que na Figura 17.7, um gr\303\241fico da fun\303\247\303\243o e de sua derivada, forem obtidos.

Usando o comando solve (resolver), encontra-se que a fun\303\247\303\243o \303\251 crescente em solve( df3 > 0, x ); e decrescente em solve( df3 < 0, x );

Figura 17.7 prov\303\252 um gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRImZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1JKG1mZW5jZWRHRiQ2JC1GIzYkLUZJNiVRInhGJ0ZMRk9GL0YvRi9GK0Yv, com intervalos crescentes mostrados em vermelho e decrescente mostrados em azul. A informa\303\247\303\243o \303\251 essencialmente a mesma que a encontrada na Figura 17.6, criada muito mais facilmente com o maplet Analizador de Fun\303\247\303\243o ( FunctionAnalyzer). p1 := plot(f3, x=-2 ..-1, color=red): p2 := plot(f3, x=-1 ..3/5, color=blue): p3 := plot(f3, x=3/5..5, color=red): p4 := plot(df3, x=-2..5, color=green): display([p1,p2,p3,p4], view=[-2..5,-100..200]); plot([0$4],x=0..1, .1..0.2, scaling =constrained, legend = ["y = f(x): increasing", "y = f(x): decreasing", "y = f(x): increasing", "y = f'(x)"], color=[red,blue,red,green]); Figura 17.7 Exemplo 17.3: O conte\303\272do da Figura 17.6, reproduzido (at length) na medida no Maple Observe como o gr\303\241fico da derivada n\303\243o muda o sinal no ponto estacion\303\241rio em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFEiM0YnRi9GL0YrRi8=. Este zero na derivada for\303\247a o gr\303\241fico da fun\303\247\303\243o a aplainar neste ponto -- mas somente neste ponto. Tecnicamente, a fun\303\247\303\243o \303\251 ainda crescente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFEiM0YnRi9GL0YrRi8= porque todos os valores da fun\303\247\303\243o \303\240 esquerda de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFEiM0YnRi9GL0YrRi8= s\303\243o negativos (negative) e todos os valores \303\240 direita s\303\243o positivos.

QUESTAO 17.1 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZWLUkjbWlHRiQ2JVElRXNzYUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSJ+RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktRiw2JVExZnVuJiMyMzE7JiMyMjc7b0YnRi9GMkY1LUYsNiVRImZGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1GNjYuRjhGUkZURjlGO0Y9Rj9GQUZDRkVGR0ZKLUkobWZlbmNlZEdGJDYlLUYjNiYtRiw2JVEieEYnRlJGVEZSLyUrZXhlY3V0YWJsZUdGMUZURlJGVEZWLUY2Ni5RIj1GJ0ZSRlRGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4yNzc3Nzc4ZW1GJy9GS0Zgb0ZWLUZZNiUtRiM2MC1GNjYuUSomdW1pbnVzMDtGJ0ZSRlRGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4yMjIyMjIyZW1GJy9GS0Zqby1JI21uR0YkNiVRIjlGJ0ZSRlRGVi1JJW1zdXBHRiQ2JUZnbi1GIzYlLUZdcDYlUSIyRidGUkZURlJGVC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGVi1GNjYuUSIrRidGUkZURjlGO0Y9Rj9GQUZDRkVGaW9GW3AtRl1wNiVRIjVGJ0ZSRlRGVkZnbkZbcS1GXXA2JVEiN0YnRlJGVEZSRmpuRlRGUkZULUYsNiVRJHNpbkYnRlJGVEZYRjUtRiw2JVEqZXN0JiMyMjU7RidGL0YyRjUtRiw2JVErYXVtZW50YW5kb0YnRi9GMkY1LUYsNiVRI291RidGL0YyRjUtRiw2JVErZGltaW51aW5kb0YnRi9GMkY1LUYsNiVRI25vRidGL0YyRjUtRiw2JVEqaW50ZXJ2YWxvRidGL0YyRjUtRiw2JVEpY29udGVuZG9GJ0YvRjJGNS1GLDYlRmluRi9GMkY1LUY2Ni1GXm9GMkY5RjtGPUY/RkFGQ0ZFRl9vRmFvRjUtRiw2JVEoYCR+Tk4yYEYnRlJGVC1JJm10ZXh0R0YkNiRRYW8/fnwrKEVzdGV+aW50ZXJ2YWxvfmlyJiMyMjU7fnNlcn5hcGVuYXN+b35hdW1lbnRvLH5vdX5hcGVuYXN+ZGltaW51aXIuKUYnRjItSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGXHQvJSpsaW5lYnJlYWtHUShuZXdsaW5lRidGNS1GLDYlUSlSZXNwb25kYUYnRi9GMkY1LUYsNiVRKGRpemVuZG9GJ0YvRjJGNS1GLDYlUSNzZUYnRi9GMkY1RmdxRjVGanFGNUZdckY1RmByLUY2Ni1RIi5GJ0YyRjlGO0Y9Rj9GQUZDRkVGR0ZKRmpuRjI= QUESTAO 17.2 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 QUESTAO17.3 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

Grosseiramente (Roughly) falando, concavidade \303\251 uma propriedade local de um gr\303\241fico que descreve a curva decrescer e ent\303\243o crescer, ou crescer e ent\303\243o decrescer, conforme a curva se desenvolve da esquerda para a direita. No primeiro caso, o gr\303\241fico ter\303\241 a forma \342\213\203 e \303\251 chamada " concavidade para cima" enquanto que no segundo caso, ter\303\241 a forma \342\213\202 e \303\251 chamada " concavidade para baixo." Defini\303\247\303\243o 17.2 faz esta no\303\247\303\243o precisa ligando a descri\303\247\303\243o ao comportamento da primeira derivada. Defini\303\247\303\243o 17.2: Concavidade Seja f uma fun\303\247\303\243o definida no intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . f \303\251 c\303\264ncava para cima em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= se f ' \303\251 crescente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ; f is c\303\264ncava para baixo em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= se f ' \303\251 decrescente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . Um ponto de inflex\303\243o de f ocorre em cada ponto onde f muda a concavidade.

Uma fun\303\247\303\243o linear (linear) n\303\243o \303\251 c\303\264ncava para cima nem c\303\264ncava para baixo (concave down). pontos de inflex\303\243o ocorrem somente em pontos do dom\303\255nio da fun\303\247\303\243o.

Na primeira parte deste cap\303\255tulo aprendemos que uma fun\303\247\303\243o deferenci\303\241vel, f, \303\251 crescente quando a sua derivada \303\251 positiva e decrescente quando sua derivada \303\251 negativa. Aplicada \303\240 primeira derivada, n\303\263s observamos que f ' \303\251 crescente quando (f ')' = f '' \303\251 positiva e que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmcHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== \303\251 decrescente 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 \303\251 negativa. Estas observa\303\247\303\265es colocadas de modo mais exato no Teorema 17.2. Teorema 17.2: Teste da Segunda-Derivada para a Concavidade Supponha ser f uma fun\303\247\303\243o que cont\303\255nua no intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e tem duas derivadas em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (exceto possivelmente nos pontos limites) Ent\303\243o, f \303\251 c\303\264ncava para cima em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= se 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 > 0 para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (exceto possivelmente nos pontos limites) f \303\251 c\303\264ncava para baixo em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Jy1JI21pR0YkNiVRJWZ+JydGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1GIzYmLUYsNi1RIihGJ0YvL0YzRk5GNS9GOEZORjlGO0Y9Rj8vRkJRLDAuMTY2NjY2N2VtRicvRkVGZm4tRiM2JC1GSTYlUSJ4RidGTEZPRi8tRiw2LVEiKUYnRi9GWUY1RlpGOUY7Rj1GP0Zlbi9GRVEsMC4xMTExMTExZW1GJ0YvRitGL0YrRi8= < 0 para todol LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (exceto possivelmente nos pontos limites)

Uma conseq\303\274\303\252ncia deste teorema \303\251 que um ponto de inflex\303\243o \303\251 um ponto onde a segunda derivada muda de sinal. Este pode ser um ponto onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2J0YrLUYjNictSSNtaUdGJDYlUSVmficnRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYsNi1RMCZBcHBseUZ1bmN0aW9uO0YnRi9GMkY1RjdGOUY7Rj1GP0ZBRkQtRiM2Ji1GLDYtUSIoRidGLy9GM0ZQRjUvRjhGUEY5RjtGPUY/L0ZCUSwwLjE2NjY2NjdlbUYnL0ZFRmhuLUYjNiQtRks2JVEieEYnRk5GUUYvLUYsNi1RIilGJ0YvRmVuRjVGZm5GOUY7Rj1GP0Znbi9GRVEsMC4xMTExMTExZW1GJ0YvRitGLy1GLDYtUSI9RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRmhvLUkjbW5HRiQ2JFEiMEYnRi9GL0YrRi8= ou um ponto onde 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 n\303\243o existe (mas est\303\241 fora do dom\303\255nio de f). Pontos onde 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 s\303\243o chamados poss\303\255veis pontos de inflex\303\243o. \303\211 necess\303\241rio checar a mudan\303\247a de sinal de f '' neste ponto. A hip\303\263tese (hypotheses) pode ser simplificada para to "Suponha ser f uma fun\303\247\303\243o que tenha uma segunda derivada no intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=." Se uma fun\303\247\303\243o tem uma segunda derivada em um intervalo, ent\303\243o a primeira derivada e a fun\303\247\303\243o devem ser ambas cont\303\255nuas. O coment\303\241rio (listing) expl\303\255cito de continuidade \303\251 um lembrete para checar esta propriedade antes de tentar aplicar o teorema.

Encontre os intervalos em que a fun\303\247\303\243o f4 := x^3-3*x^2-x: f(x) = f4; \303\251 c\303\264ncava para baixo e c\303\264ncava para cima; identifique quaisquer pontos de inflex\303\243o. Figura 17.8 mostra a janela do gr\303\241fico que resulta quando o maplet Analisador de Fun\303\247\303\243o (FunctionAnalyzer) \303\251 aplicado a 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 A curva vermelha \303\251 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 a reta azul , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZQcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGL0YyRjlGOS1GNjYtUSIuRidGOUY7Rj5GQEZCRkRGRkZIL0ZLRk9GTUY5 O sinal indicador usa o marrom para indicar 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 e verde para indicar 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 Portanto, onde o sinal indicador \303\251 marrom, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2JC1JI21pR0YkNiVRImZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGL0YrRi8= \303\251 c\303\264ncava para baixo, e onde \303\251 verde, c\303\264ncava para cima. 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Figura 17.8 Exemplo 17.4: Janela gr\303\241fica do maplet FunctionAnalyzer aplicada a 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 A abordagem anal\303\255tica usa os pontos singulares e poss\303\255veis pontos de inflex\303\243o (solu\303\247\303\265es para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2J0YrLUYjNictSSNtaUdGJDYlUSVmficnRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYsNi1RMCZBcHBseUZ1bmN0aW9uO0YnRi9GMkY1RjdGOUY7Rj1GP0ZBRkQtRiM2Ji1GLDYtUSIoRidGLy9GM0ZQRjUvRjhGUEY5RjtGPUY/L0ZCUSwwLjE2NjY2NjdlbUYnL0ZFRmhuLUYjNiQtRks2JVEieEYnRk5GUUYvLUYsNi1RIilGJ0YvRmVuRjVGZm5GOUY7Rj1GP0Znbi9GRVEsMC4xMTExMTExZW1GJ0YvRitGLy1GLDYtUSI9RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRmhvLUkjbW5HRiQ2JFEiMEYnRi9GL0YrRi8= ) para dividir o dom\303\255nio em intervalos onde a segunda derivada ser\303\241 ou positiva ou negativa em todo intervalo. Esta fun\303\247\303\243o \303\251 a mesma do Exemplo 17.1. E mais uma vez, porque a fun\303\247\303\243o \303\251 polinomial, n\303\243o existe singularidades para esta fun\303\247\303\243o, ou para quaisquer de suas derivadas. A segunda derivada \303\251 ddf4 := diff( f4, x,x ): `f ''`(x) = ddf4; e os poss\303\255veis pontos de inflex\303\243o s\303\243o as solu\303\247\303\265es de 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, que s\303\243o justamente possinfl4 := solve( ddf4 = 0, x ): possinfl4; Isto significa que a segunda derivada ter\303\241 um sinal em cada um dos dois intervalos seguintes 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 Para determinar se a fun\303\247\303\243o \303\251 c\303\264ncava para cima ou c\303\264ncava para baixo em um intervalo, simplesmente avalie a derivada em um ponto conveniente em cada subintervalo. O comando TestConcavity (Teste de Concavidade), definido na se\303\247\303\243o inicializa\303\247\303\243o, avalia a segunda derivada em cada ponto e usa o sinal do resultado para informar se a fun\303\247\303\243o \303\251 c\303\264ncava para cima ou c\303\264ncava para baixo ou se existe um poss\303\255vel ponto de inflex\303\243o. testpts4 := [ 0, 2 ]: TestConcavity( ddf4, testpts4 ); Portanto, a fun\303\247\303\243o 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 \303\251 c\303\264ncava para cima 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 e c\303\264ncava para baixo em 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.

Using the solve command, the function is found to be concave up on solve( ddf4 > 0, x ); and concave down on solve( ddf4 < 0, x );

Figura 17.9 prov\303\252 um gr\303\241fico desta fun\303\247\303\243o e de sua segunda derivada. O intervalo da concavidade para cima interval \303\251 mostrado em rosa; o intervalo da concavidade para baixo, em azul piscina (verde azulado). O conte\303\272do da Figura 17.9 \303\251 essencialmente o mesmo que o da Figura 17.8. p1 := plot( f4, x=-2..1, color=pink, legend=["y = f(x): concave up"] ): p2 := plot( f4, x= 1..4, color=cyan, legend=["y = f(x): concave down"] ): p3 := plot( ddf4, x=-2..4, color=turquoise, legend=["y = f''(x)"] ): display( [p1,p2,p3], thickness=3 ); Figura 17.9 Conte\303\272do da Figura 17.8 gerado diretamente no Maple Observe como o gr\303\241fico da segunda derivada (linha azul) muda de sinal justamente onde a fun\303\247\303\243o muda a concavidade, em t LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 1.

Os tr\303\252s maplets \303\272teis para determinar intervalos onde uma fun\303\247\303\243o \303\251 crescente e decrescente s\303\243o tamb\303\251m \303\272teis para determinar onde uma fun\303\247\303\243o \303\251 c\303\264ncava para cima e para baixo.

O maplet An\303\241lise de |Curva ( CurveAnalysis maplet) cont\303\251m bot\303\265es para computar intervalos onde a fun\303\247\303\243o \303\251 c\303\264ncava para cima e onde \303\251 c\303\264ncava para baixo. Da mesma forma, a informa\303\247\303\243o da concavidade pode ser exibido no gr\303\241fico. O uso \303\251 completamente paralelo ao uso deste maplet para an\303\241lise de crescente/decrescente. Figura 17.10 mostra como este maplet (ou eq\303\274ivalentemente, inserir no Tutor de An\303\241lise de Curva (Curve Analysis Tutor)) pareceria se fosse usado para achar onde a fun\303\247\303\243o seria c\303\264ncava para cima. Figura 17.10 Exemplo 17.4: Tutor de An\303\241lise de curva (Curve Analysis Tutor) usado para determinar onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjZRJ25vcm1hbEYn \303\251 c\303\264ncava para cima 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Figura 17.10 Exemplo 17.4: Tutor de An\303\241lise de Curva (Curve Analysis Tutor) usado para determinar onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 c\303\264ncava para cima

O maplet Analisador de Fun\303\247\303\243o (FunctionAnalyzer maplet) \303\251 similar. Use o painel Segunda Derivada para especificar se o gr\303\241fico deve incluir a segunda derivada e/ou o sinal indicador para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ==. Para especificar a segunda derivada na equa\303\247\303\243o, use D2y. Por exemplo, para encontrar o(s) intervalo(s) onde a fun\303\247\303\243o \303\251 c\303\264ncava para baixo, entre com D2y < 0 no campo denominado (Enter equation) Entre com equa\303\247\303\243o para ser resolvida (use y, Dy, D2y). A Figura 17.8 d\303\241 uma pr\303\251via de como este maplet would exibe informa\303\247\303\243o sobre concavidade.

O maplet Gr\303\241fico da Derivada (DerivativePlot maplet) (o inserido em Tutor de Derivada (Derivative Tutor)) n\303\243o requer nenhuma discuss\303\243o adicional exceto para dizer que o gr\303\241fico da derivada ser\303\241 inclu\303\255do no gr\303\241fico se a caixa da segunda derivada for clicado. Figura 17.11 mostra como este maplet exibe informa\303\247\303\243o sobre a concavidade da fun\303\247\303\243o. 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Figura 17.11 Exemplo 17.4: Use do Tutor de Derivada (Derivative Tutor) para exibir a concavidade

Encontre os intervalos em que a fun\303\247\303\243o f5 := (x^3-3*x^2-x)/(x^2-4): f(x) = f5; \303\251 c\303\264ncava para cima e c\303\264ncava para baixo; identifique quaisquer pontos de inflex\303\243o. A Figura 17.12 mostra a janela do gr\303\241fico que resulta da aplica\303\247\303\243o o maplet Analisador de Fun\303\247\303\243o (FunctionAnalyzer) para 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Figura 17.12 Exemplo 17.5: Janela gr\303\241fica do Analisador de Fun\303\247\303\243o (FunctionAnalyzer) Figura 17.12 Exemplo 17.5: Graphing window from FunctionAnalyzer maplet O gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 dado em azul, mas as duas retas azuis "verticais" n\303\243o s\303\243o realmente ass\303\255ntotas. Elas s\303\243o retas esp\303\272rias geradas pelo algor\303\255tmo gr\303\241fico conectar-os-pontos do Maple. O maplet Analisador de Fun\303\247\303\243o (FunctionAnalyzer) n\303\243o faz previs\303\243o de suprimir estas liga\303\247\303\265es de segmentos cont\303\255nuos de um gr\303\241fico. A caracter\303\255stica \303\272til da Figura 17.12 \303\251 o sinal indicador para a segunda derivada. Onde ele \303\251 vermelho, a fun\303\247\303\243o \303\251 c\303\264ncava para baixo; onde \303\251 verde, para cima. Seria muito dif\303\255cil detectar a mudan\303\247a na concavidade que ocorre \303\240 direita de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzEwRidGOUY5, como indicado pelo sinal. Os c\303\241lculos abaixo mostrar\303\243o que o sinal indicador est\303\241 realmente correto! Esta fun\303\247\303\243o \303\251 a mesma do Exemplo 17.2. Lembre-se de que o dom\303\255nio desta fun\303\247\303\243o \303\251(?)todos os n\303\272meros reais exceto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFEiMkYnRi9GL0YrRi8= e 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 . As duas primeiras derivadas s\303\243o df5 := simplify( diff( f5, x ) ): ddf5 := simplify( diff( f5, x,x ) ): `f '`(x) = df5; `f ''`(x) = ddf5; Estas fun\303\247\303\265es s\303\243o claramente definidas e cont\303\255nuas no dom\303\255niode f, i.e., todos os n\303\272meros LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= reais exceto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFEiMkYnRi9GL0YrRi8= e 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 . Porque a fun\303\247\303\243o \303\251 descont\303\255nua (discontinuous) nestes pontos de singularidade, sing5 := -2, 2: sing5; \303\211 poss\303\255vel a concavidade mudar nestes pontos. Estes pontos ser\303\243o limites para intervalos onde a segunda derivada n\303\243o mudam o sinal. Dos tr\303\252s zeros do numerador da segunda derivada, existe somente uma solu\303\247\303\243o real, a saber, q1 := solve( numer(ddf5) = 0, x ): possinfl5 := op(remove( has, [q1], I )): possinfl5 = evalf[4]( possinfl5 ); Este \303\251 o \303\272nico ponto de inflex\303\243o poss\303\255vel porque 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s\303\243o pontos de singularidade e n\303\243o est\303\243o no dom\303\255nio da fun\303\247\303\243o. A segunda derivada pode mudar de sinal em um ponto onde a segunda derivada \303\251 zero (um poss\303\255vel ponto de inflex\303\243o) ou onde a fun\303\247\303\243o e/ou sua segunda derivada n\303\243o est\303\241 definida (um ponto singular). Portanto, a reta real \303\251 dividida nos quatro intervalos seguintes em que a fun\303\247\303\243o n\303\243o muda a concavidade 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 Para determinar em quais destes intervalos a fun\303\247\303\243o \303\251 crescente e em quais \303\251 decrescente, simplesmente avalie a derivada em um ponto conveniente em cada subintervalo. testpts5 := [ -3, 0, 3, 12 ]: TestConcavity( ddf5, testpts5 ); Portanto, a fun\303\247\303\243o 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 \303\251 c\303\264ncava oara cima emLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21uR0YkNiRRKyZ1bWludXMwOzJGJ0YvLUYsNi1RIixGJ0YvRjIvRjZRJXRydWVGJ0Y3RjlGO0Y9Rj9GQS9GRVEsMC4zMzMzMzMzZW1GJy1GTDYkUSIyRidGL0YvRi9GK0Yv e 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 e c\303\264ncava para baixo 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 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21uR0YkNiRRIjJGJ0YvLUYsNi1RIixGJ0YvRjIvRjZRJXRydWVGJ0Y3RjlGO0Y9Rj9GQS9GRVEsMC4zMzMzMzMzZW1GJy1GTDYkUSYxMS4wNEYnRi9GL0YvRitGLw==. Porque um ponto de inflex\303\243o deve, por defini\303\247\303\243o, ocorrer em um ponto no dom\303\255nio da fun\303\247\303\243o, esta fun\303\247\303\243o tem somente um ponto de inflex\303\243o, em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFEmMTEuMDRGJ0YvRi9GK0Yv .

Using the solve command, the function is found to be concave up on solve( ddf5 > 0, x ); and concave down on solve( ddf5 < 0, x );

A Figura 17.13 prov\303\252 um gr\303\241fico da fun\303\247\303\243o e sua segunda derivada (em verde). Os intervalos de concavidade para cima s\303\243o mostrados em preto; os intervalos de concavidade para baixo, em vermelho. p1 := plot( f5, x=-5 ..-2, color=red): p2 := plot( f5, x=-2 ..2, color=black): p3 := plot( f5, x= 2 ..11.04, color=red): p4 := plot( f5, x=11.04..16, color=black): p5 := plot(ddf5, x=-6 ..26, color=green, discont=true, thickness=2): display([p||(1..5)], view=[-6..16,-30..30]); plot([0$5],x=0..1, .1..0.2, scaling =constrained, legend = ["f(x): concave down", "f(x): concave up", "f(x): concave down", "f(x): concave up", "f ''(x)"], color=[red,black,red,black,green]); Figura 17.13 Exemplo 17.5: Gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Este gr\303\241fico n\303\243o \303\251 t\303\243o claro como se esperava. Nesta escala, \303\251 imposs\303\255vel ver que realmente existe um ponto de inflex\303\243o. A Figura 17.14 prov\303\252 uma vis\303\243o ampliada da segunda derivada. plot(ddf5, x=10..20, color=green); Figura 17.14 Exemplo 17.5: Amplia\303\247\303\243o (Zooming) sobre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Note a escala sobre o eixo vertical (vertical axis). Como a Figura 17.14 revela a mudan\303\247a de sinal na segunda derivada, nossa confian\303\247a na an\303\241lise foi restaurada!

Encontre os intervalos em que a fun\303\247\303\243o f6 := x^5-7*x^4+10*x^3+18*x^2-27*x-27: f(x) = f6; \303\251 c\303\264ncava para baixo; identifique todos pontos de inflex\303\243o A Figure 17.15 mostra a janela do gr\303\241fico que resulta quando o maplet Analisador de Fun\303\247\303\243o (FunctionAnalyzer) \303\251 aplicado \303\240 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk0tRjY2LVEiLkYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GOQ== A fun\303\247\303\243o \303\251 grafada em marrom, a segunda derivada em verde. Onde o sinal indicador \303\251 vermelho, a fun\303\247\303\243o \303\251 c\303\264ncava para baixo; onde \303\251 azul, para cima. 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Figura 17.15 Exemplo 17.6: Aplica\303\247\303\243o do maplet Analisador de Fun\303\247\303\243o (FunctionAnalyzer), usando marrom para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2JC1JI21pR0YkNiVRImZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGL0YrRi8= e verde para 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. O sinal indicador para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== usa vermelho para negativo e azul para positivo. N\303\243o existe singularidades; somente os poss\303\255veis pontos de inflex\303\265es precisam ser encontrados. A segunda derivada \303\251 um polin\303\264mio do terceiro grau ddf6 := diff( f6, x,x ): `f''`(x) = ddf6; cujos zeros s\303\243o possinfl6 := solve( ddf6 = 0, x ): possinfl6; evalf[4](possinfl6); Como a derivada \303\251 cont\303\255nua, o Teorema do Valor da M\303\251dia ( Intermediate Value Theorem) permite concluir que a segunda derivada ter\303\241 um sinal em cada um dos quatros intervalos seguintes 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 Para determinar em quais destes intervalos a fun\303\247\303\243o \303\251 c\303\264ncava para cima e em quais \303\251 c\303\264ncava para baixo, simplesmente avalie a segunda derivada em um ponto conveniente em cada subintervalo. testpts6 := [ -1, 0, 2, 4 ]: TestConcavity( ddf6, testpts6 ); Portanto, a fun\303\247\303\243o 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 \303\251 c\303\264ncava para cima emLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21uR0YkNiRRLiZ1bWludXMwOzAuMzhGJ0YvLUYsNi1RIixGJ0YvRjIvRjZRJXRydWVGJ0Y3RjlGO0Y9Rj9GQS9GRVEsMC4zMzMzMzMzZW1GJy1GTDYkUSUxLjU4RidGL0YvRi9GK0Yv e 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 e c\303\264ncava para baixo em 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 eLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21uR0YkNiRRJTEuNThGJ0YvLUYsNi1RIixGJ0YvRjIvRjZRJXRydWVGJ0Y3RjlGO0Y9Rj9GQS9GRVEsMC4zMzMzMzMzZW1GJy1GTDYkUSIzRidGL0YvRi9GK0Yv. Os tr\303\252s pontos de inflex\303\243o ocorrem em 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFElMS41OEYnRi9GL0YrRi8= , e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFEiM0YnRi9GL0YrRi8=.

Using the solve command, the function is found to be increasing on solve( ddf6 > 0, x ); and decreasing on solve( ddf6 < 0, x );

A Figura 17.16 prov\303\252 um gr\303\241fico da fun\303\247\303\243o e da sua segunda derivada (em verde). Os intervalos de concavidade para cima s\303\243o mostrados em preto; e os intervalos de concavidade para baixo, em vermelho. p1 := plot( f6, x=-2 ..-0.38, color=red): p2 := plot( f6, x=-0.38.. 1.58, color=black): p3 := plot( f6, x= 1.58.. 3, color=red): p4 := plot( f6, x= 3 .. 5, color=black): p5 := plot( ddf6, x=-3..5, color=green): display([p||(1..5)], view=[-2..5,-100..200]); plot([0$5],x=0..1, .1..0.2, scaling =constrained, legend = ["f(x): concave down", "f(x): concave up","f(x): concave down", "f(x): concave up", "f ''(x)"], color=[red,black,red,black,green]); Figura 17.16 Exemplo 17.6: Gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Os tr\303\252s pontos de inflex\303\243o s\303\243o claramente vis\303\255veis neste gr\303\241fico. Deve ser notado tamb\303\251m que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Ji1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZVLUkjbW5HRiQ2JFEiM0YnRi9GL0YrRi8= \303\251 um ponto de inflex\303\243o e tamb\303\251m \303\251 um ponto estacion\303\241rio.

Durante uma caminhada pelas montanhas, algu\303\251m certamente se interessaria em saber onde est\303\241 a montanha mais alta, e onde deve estar o vale mais profundo. E mais, poderia tamb\303\251m ser de interesse saber onde est\303\243o localizados os outros picos e as outras depress\303\265es. justamente como o mais alto e o mais baixo de um ranque coresponde a extremo (absoluto) global, as localiza\303\247\303\265es dos picos menores e as outras depress\303\265es correspodem ao extremo local (ou relativo). Este novo conceito \303\251 colocado de modo preciso na Defini\303\247\303\243o 17.3 Defini\303\247\303\243o 17.3: Local de Extremo Seja f uma fun\303\247\303\243o com dom\303\255nio domain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Um ponto c em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 chamado de um m\303\255nimo local de f se em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= existe um intervalo limitado e fechado , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, que contenha LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e para o qual o m\303\255nimo global de f em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ocorre em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= um m\303\241ximo local de f se em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= existe um intervalo limitado e fechado, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, que contenha LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e para o qual o m\303\241ximo global de f em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ocorre em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

Observe como os locais de extremo s\303\243o definidos em termos de extremo global em um intervalo apropriado limitado e fechado. Se c \303\251 um local de m\303\255nimo de f, ent\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Jy1JI21pR0YkNiVRImZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1GIzYmLUYsNi1RIihGJ0YvL0YzRk5GNS9GOEZORjlGO0Y9Rj8vRkJRLDAuMTY2NjY2N2VtRicvRkVGZm4tRiM2JC1GSTYlUSJjRidGTEZPRi8tRiw2LVEiKUYnRi9GWUY1RlpGOUY7Rj1GP0Zlbi9GRVEsMC4xMTExMTExZW1GJ0YvRitGL0YrRi8= \303\251 o valor de f correspondente ao m\303\255nimo local . Se c \303\251 um local de m\303\241ximo de f, ent\303\243o 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 \303\251 o valor de f correspondente ao m\303\241ximo local .