Chapter 3 - Applications of Differentiation - 3.3 Exercises - Page 183: 39

\[\begin{align} & \left( \text{a} \right)\text{ }x=1 \\ & \left( \text{b} \right)\text{Increasing on}\left( -\infty ,1 \right),\text{ decreasing on }\left( 1,\infty \right) \\ & \left( \text{c} \right)\text{Relative maximum: }\left( 1,4 \right) \\ \end{align}\]

\[\begin{align} & \text{Let }f\left( x \right)=\left\{ \begin{matrix} 3x+1,\text{ }x\le 1 \\ 5-{{x}^{2}},\text{ }x>1 \\ \end{matrix} \right. \\ & \left( \text{a} \right) \\ & \text{Differentiate } \\ & f'\left( x \right)=\left\{ \begin{matrix} \frac{d}{dx}\left[ 3x+1 \right],\text{ }x\le 1 \\ \frac{d}{dx}\left[ 5-{{x}^{2}} \right],\text{ }x>1 \\ \end{matrix} \right. \\ & f'\left( x \right)=\left\{ \begin{matrix} 3,\text{ }x\le 1 \\ -2x,\text{ }x>1 \\ \end{matrix} \right. \\ & \text{We obtain the critical point }x=1\text{ }\left( \text{discontinuity} \right) \\ & \text{Set the intervals }\left( -\infty ,1 \right),\left( 1,\infty \right) \\ & \\ & \left( \text{b} \right)\text{Making a table of values } \\ & \begin{matrix} \text{Interval} & \left( -\infty ,1 \right) & \left( 1,\infty \right) \\ \text{Test Value} & x=0 & x=1 \\ \text{Sign of }f'\left( x \right) & \text{ }f'\left( 0 \right)=3>0 & \text{ }f'\left( 2 \right)=-2<0 \\ \text{Conclusion} & \text{Increasing} & \text{Decreasing} \\ \end{matrix} \\ & \\ & \text{By Theorem 3}\text{.6} \\ & f'\left( x \right)\text{ changes from positive to negative at }x=1,\text{ then }f\left( x \right) \\ & \text{has a relative maximum at }\left( 1,f\left( 1 \right) \right) \\ & f\left( x \right)=\left\{ \begin{matrix} 3x+1,\text{ }x\le 1 \\ 5-{{x}^{2}},\text{ }x>1 \\ \end{matrix} \right. \\ & f\left( 1 \right)=3\left( 1 \right)+1 \\ & f\left( 1 \right)=4 \\ & \text{Relative maximum: }\left( 1,4 \right) \\ \end{align}\]