Chapter 4 - Applications of the Derivative - 4.2 What Derivatives Tell Us - 4.2 Exercises - Page 256: 24

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

\[\begin{align} & f\left( x \right)=2{{x}^{5}}-\frac{15{{x}^{4}}}{4}+\frac{5{{x}^{3}}}{3} \\ & \text{Diferentiate } \\ & f\left( x \right)=10{{x}^{4}}-\frac{4\left( 15 \right){{x}^{3}}}{4}+\frac{3\left( 5 \right){{x}^{2}}}{3} \\ & f'\left( x \right)=10{{x}^{4}}-15{{x}^{3}}+5{{x}^{2}} \\ & \text{Calculate the critical points, set }f'\left( x \right)=0 \\ & 10{{x}^{4}}-15{{x}^{3}}+5{{x}^{2}}=0 \\ & \text{Factoring} \\ & 5{{x}^{2}}\left( 2{{x}^{2}}-3x+1 \right)=0 \\ & 5{{x}^{2}}\left( 2x-1 \right)\left( x-1 \right)=0 \\ & \text{We obtain} \\ & x=0,\text{ }x=\frac{1}{2},\text{ }x=1 \\ & \text{From the critical values we can make the following intervals}\, \\ & \left( -\infty ,0 \right),\left( 0,\frac{1}{2} \right),\left( \frac{1}{2},1 \right),\left( 1,\infty \right) \\ & \text{Now, we will evaluate between the critical values and resume } \\ & \text{in a table} \\ & \text{ }\begin{matrix} \text{Interval} & \text{Test value}\left( x \right) & \text{Sign of }{f}'\left( x \right) & \text{Behavior of }f\left( x \right) \\ \left( -\infty ,0 \right) & -1 & + & \text{Increasing} \\ \left( 0,\frac{1}{2} \right) & 0.25 & + & \text{Increasing} \\ \left( \frac{1}{2},1 \right) & 0.75 & - & \text{Decreasing} \\ \left( 1,\infty \right) & 2 & + & \text{Increasing} \\ {} & {} & {} & {} \\ {} & {} & {} & {} \\ \end{matrix} \\ & \text{From the table we can conlude that the function is:} \\ & \text{Decreasing on }\left( \frac{1}{2},1 \right) \\ & \text{Increasing on }\left( -\infty ,\frac{1}{2} \right),\left( 1,\infty \right) \\ & \text{Graph} \\ \end{align}\]