Question

considere a função F(x)=x³/3-x²-8x+26/3 A
) Determine os pontos de máximo ou mínimo da função, se existirem. B
) Especifique qual dos valores encontrados no item anterior é ponto de máximo e qual é ponto de mínimo. C
) Determine o ponto de inflexão do gráfico da função.

Answer 1

Boa tarde Mary!

Solução!

$f(x)= \dfrac{ x^{3} }{3}- x^{2} -8x+ \dfrac{26}{3}$

Vamos derivar a função!

$f(x)= \dfrac{ x^{3} }{3}- x^{2} -8x+ \dfrac{26}{3} \\\\\\ f'(x)= \dfrac{ 3x^{(3-1)} }{3}- 2x^{(2-1)} -8 x^{(1-1)} + \dfrac{26}{3} \\\\\\\\ f'(x)= \dfrac{ 3x^{2} }{3}- 2x -8 \\\\\\\\ f'(x)= x^{2}- 2x -8$

Apos a função derivada encontramos uma equação do segundo grau,vamos determinar suas raizes.

$f(x)=0\\\\\\x^{2}- 2x -8=0\\\\\\\\ Formula ~~de~~Bhaskara!\\\\\\ x= \dfrac{-b\pm \sqrt{b^{2} -4.a.c} }{2.a}\\\\\\\ x= \dfrac{-(-2)\pm \sqrt{(-2)^{2} -4.1.(-8)} }{2.1}\\\\\\\ x= \dfrac{2\pm \sqrt{4+32} }{2}\\\\\\\ x= \dfrac{2\pm \sqrt{36} }{2}\\\\\\\ x= \dfrac{2\pm 6 }{2}\\\\\\\ x_{1}= \dfrac{2+6}{2} = \dfrac{8}{2} =4\\\\\\\ x_{2}= \dfrac{2-6}{2} = \dfrac{-4}{2} =-2\\\\\\\ \boxed{Raizes:x_{2}= -2~~x_{1}=4}$

Conhecendo os valores das raizes,vamos substituir na equação original para determinarmos o ponto de máximo e minimo.

$f(x)=y\\\\\\ y= \dfrac{ (-2)^{3} }{3}- (-2)^{2} -8(-2)+ \dfrac{26}{3}\\\\\\\\ y= \dfrac{-8 }{3}- 4 +16+ \dfrac{26}{3}\\\\\\\\ y= \dfrac{-8-12+48+26 }{3}\\\\\\\\ y= \dfrac{-8-12+48+26 }{3}\\\\\\\\ y= \dfrac{-20+74 }{3}\\\\\\\\ y= \dfrac{54 }{3}\\\\\\\\ y=18~~\Rightarrow~~Maximo$

$y= \dfrac{ (4)^{3} }{3}- (4)^{2} -8(4)+ \dfrac{26}{3}\\\\\\\\ y= \dfrac{ 64 }{3}- 16 -32+ \dfrac{26}{3}\\\\\\\\ y= \dfrac{ 64-48-96+26}{3}\\\\\\\\ y= \dfrac{90-144 }{3}\\\\\\\\ y= \dfrac{-54}{3}\\\\\\\\ y= -18~~\Rightarrow~~Minimo$

$\boxed{Ponto_{Maximo}(-2,18)~~Ponto_{Minimo}(4,-18)}$

O ponto de inflexão e obtido fazendo a segunda derivada!

$f'(x)= x^{2}- 2x -8 \\\\\\\\ f''(x)= 2x^{(2-1)}- 2x -8 \\\\\\\\ f'(x)= 2x-2\\\\\\\\ f'(x)= 0\\\\\\\ 2x-2=0\\\\\\ 2x=2\\\\\\ x= \dfrac{2}{2} \\\\\\ x=1\\\\\\ x_{inflex\~ao}=1}\\\\\ \boxed{Ponto~~de_{inflex\~ao}(1,0)}$



$\boxed{Resposta:A~~Maximo=18~~Minimo=-18}\\\\\\\ \boxed{Resposta:B~~Ponto_{Maximo}(-2,18)~~Ponto_{Minimo}(4,-18) }\\\\\\\ \boxed{Resposta:C~~x_{inflex\~ao}=1 ~~Ponto~~de_{inflex\~ao}(1,0)}}$





Boa noite!
Bons estudos!