Chapter 4 - Applications of the Derivative - 4.2 What Derivatives Tell Us - 4.2 Exercises - Page 256: 25

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

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