Chapter 4 - Applications of the Derivative - 4.2 What Derivatives Tell Us - 4.2 Exercises - Page 256: 23

\[\begin{align} & \text{Decreasing on }\left( 0,1 \right),\left( 2,\infty \right) \\ & \text{Increasing on }\left( -\infty ,0 \right),\left( 1,2 \right) \\ \end{align}\]

\[\begin{align} & f\left( x \right)=-\frac{{{x}^{4}}}{4}+{{x}^{3}}-{{x}^{2}} \\ & \text{Diferentiate } \\ & f'\left( x \right)=-\frac{4{{x}^{3}}}{4}+3{{x}^{2}}-2x \\ & f'\left( x \right)=-{{x}^{3}}+3{{x}^{2}}-2x \\ & \text{Calculate the critical points, set }f'\left( x \right)=0 \\ & -{{x}^{3}}+3{{x}^{2}}-2x=0 \\ & \text{Factoring} \\ & -x\left( {{x}^{2}}-3x+2 \right)=0 \\ & -x\left( x-2 \right)\left( x-1 \right)=0 \\ & \text{We obtain} \\ & x=0,\text{ }x=1,\text{ }x=2 \\ & \text{From the critical values we can make the following intervals}\, \\ & \left( -\infty ,0 \right),\left( 0,1 \right),\left( 1,2 \right),\left( 2,\infty \right) \\ & \text{Now, we will evaluate between the critical values and resume } \\ & \text{in a table} \\ & \text{ }\begin{matrix} \text{Interval} & \text{Test value}\left( x \right) & \text{Sign of }{f}'\left( x \right) & \text{Behavior of }f\left( x \right) \\ \left( -\infty ,0 \right) & -1 & + & \text{Increasing} \\ \left( 0,1 \right) & 0.5 & - & \text{Decreasing} \\ \left( 1,2 \right) & 1.5 & + & \text{Increasing} \\ \left( 2,\infty \right) & 3 & - & \text{Decreasing} \\ {} & {} & {} & {} \\ {} & {} & {} & {} \\ \end{matrix} \\ & \text{From the table we can conlude that the function is:} \\ & \text{Decreasing on }\left( 0,1 \right),\left( 2,\infty \right) \\ & \text{Increasing on }\left( -\infty ,0 \right),\left( 1,2 \right) \\ & \text{Graph} \\ \end{align}\]