Chapter 3 - Applications of Differentiation - 3.3 Exercises - Page 183: 32

\[\begin{align} & \left( \text{a} \right)\text{ }x=-3 \\ & \left( \text{b} \right)\text{Increasing on}\left( -3,\infty \right),\text{ decreasing on }\left( -\infty ,3 \right) \\ & \left( \text{c} \right)\text{Relative minimum: }\left( -3,-1 \right) \\ \end{align}\]

\[\begin{align} & f\left( x \right)=\left| x+3 \right|-1 \\ & \left( \text{a} \right) \\ & \text{Differentiating} \\ & f'\left( x \right)=-\frac{x+3}{\left| x+3 \right|} \\ & \text{Using the definition of the absolute value}\text{, we can write} \\ & \text{the derivative of the function as} \\ & f'\left( x \right)=\left\{ \begin{matrix} -1,\text{ }x<-3 \\ 1,\text{ }x>-3 \\ \end{matrix} \right. \\ & \text{The derivative is not defined at }x=-3,\text{ we obtain the critical } \\ & \text{point }x=-3 \\ & \text{Set the intervals }\left( -\infty ,-3 \right),\left( -3,\infty \right) \\ & \\ & \left( \text{b} \right)\text{Making a table of values }\left( \text{See examples on page 180 } \right) \\ & \begin{matrix} \text{Interval} & \left( -\infty ,-3 \right) & \left( -3,\infty \right) \\ \text{Test Value} & x=-4 & x=0 \\ \text{Sign of }f'\left( x \right) & \text{ }f'\left( -4 \right)=-1<0 & \text{ }f'\left( 0 \right)=1>0 \\ \text{Conclusion} & \text{Decreasing} & \text{Increasing} \\ \end{matrix} \\ & \\ & \left( \text{c} \right)\text{By Theorem 3}\text{.6} \\ & f'\left( x \right)\text{ changes from negative to positive at }x=-3,\text{ then }f\left( x \right) \\ & \text{has a relative minimum at }\left( -3,f\left( -3 \right) \right) \\ & f\left( -3 \right)=\left| -3+3 \right|-1 \\ & f\left( -3 \right)=-1 \\ & \text{Relative minimum: }\left( -3,-1 \right) \\ \end{align}\]