Chapter 4 - Applications of the Derivative - Review Exercises - Page 330: 7

\[\begin{align} & \text{Critical points: }x=-2,\text{ }x=3 \\ & \text{no absolute max or minimum} \\ \end{align}\]

\[\begin{align} & f\left( x \right)=2{{x}^{3}}-3{{x}^{2}}-36x+12\text{ on }\left( -\infty ,\infty \right) \\ & \text{Differentiate} \\ & f'\left( x \right)=6{{x}^{2}}-6x-36 \\ & \text{Calculate the critial points, set }f'\left( x \right)=0 \\ & 6{{x}^{2}}-6x-36=0 \\ & {{x}^{2}}-x-6=0 \\ & \left( x-3 \right)\left( x+2 \right)=0 \\ & \text{Therefore} \\ & \text{We have the critical points} \\ & x=-2,3 \\ & \text{Evaluating }f\text{ at each of these points, we have} \\ & f\left( -2 \right)=2{{\left( -2 \right)}^{3}}-3{{\left( -2 \right)}^{2}}-36\left( -2 \right)+12 \\ & f\left( -2 \right)=56 \\ & f\left( 3 \right)=2{{\left( 3 \right)}^{3}}-3{{\left( 3 \right)}^{2}}-36\left( 3 \right)+12 \\ & f\left( 3 \right)=-69 \\ & \text{The largest of these function values is:} \\ & f\left( -2 \right)=56 \\ & \text{Which is the local maximum on }\left( -\infty ,\infty \right) \\ & \text{The smallest of those function values are:} \\ & f\left( 3 \right)=-69 \\ & \text{Which is the local minimum on }\left( -\infty ,\infty \right) \\ & \text{There are no absolute max or minimum} \\ & \\ & \text{Critical points: }x=-2,\text{ }x=3 \\ & \text{no absolute max or minimum} \\ & \text{Graph} \\ \end{align}\]