Chapter 3 - Applications of Differentiation - 3.3 Exercises - Page 183: 31

$$\eqalign{ & \left( {\text{a}} \right){\text{ }}x = 5 \cr & \left( {\text{b}} \right){\text{Increasing on}}\left( { - \infty ,5} \right),{\text{ decreasing on }}\left( {5,\infty } \right) \cr & \left( {\text{c}} \right){\text{Relative maximum: }}\left( {5,5} \right) \cr} $$

\[\begin{align} & f\left( x \right)=5-\left| x-5 \right| \\ & \left( \text{a} \right) \\ & \text{Differentiating} \\ & f\left( x \right)=-\frac{x-5}{\left| x-5 \right|},\text{ or} \\ & \text{Using the definition of the absolute value}\text{, we can write} \\ & \text{the function as} \\ & f\left( x \right)=\left\{ \begin{matrix} x,\text{ }x<5 \\ -x+10,\text{ }x>5 \\ \end{matrix} \right. \\ & \text{Differentiating} \\ & f'\left( x \right)=\left\{ \begin{matrix} \frac{d}{dx}\left( x \right),\text{ }x<5 \\ \frac{d}{dx}\left( -x+10 \right),\text{ }x>5 \\ \end{matrix} \right. \\ & f'\left( x \right)=\left\{ \begin{matrix} 1,\text{ }x<5 \\ -1,\text{ }x>5 \\ \end{matrix} \right. \\ & \text{The derivative is not defined at }x=5,\text{ we obtain the critical } \\ & \text{point }x=5 \\ & \text{Set the intervals }\left( -\infty ,5 \right),\left( 5,\infty \right) \\ & \\ & \text{Making a table of values: }\\ & \begin{matrix} \text{Interval} & \left( -\infty ,5 \right) & \left( 5,\infty \right) \\ \text{Test Value} & x=0 & x=6 \\ \text{Sign of }f'\left( x \right) & \text{ }f'\left( 0 \right)=1>0 & \text{ }f'\left( 1 \right)=-1<0 \\ \text{Conclusion} & \text{Increasing} & \text{Decreasing} \\ \end{matrix} \\ & \\ & \text{By Theorem 3}\text{.6} \\ & f'\left( x \right)\text{ changes from positive to negative at }x=5,\text{ then }f\left( x \right) \\ & \text{has a relative maximum at }\left( 5,f\left( 5 \right) \right) \\ & f\left( 5 \right)=5-\left| 5-5 \right| \\ & f\left( 5 \right)=5 \\ & \text{Relative maximum: }\left( 5,5 \right) \\ \end{align}\]