Chapter 3 - Applications of Differentiation - 3.3 Exercises - Page 183: 40

\[\begin{align} & \left( \text{a} \right)x=0,\text{ }x=1 \\ & \left( \text{b} \right)\text{Decreasing on: }\left( -\infty ,0 \right)\text{ and }\left( 1,\infty \right)\text{ } \\ & \text{Increasing on: }\left( 0,1 \right),\text{ } \\ & \left( \text{c} \right)\text{relative maximum}\left( 1,1 \right) \\ & \text{ relative minimum}\left( 0,1 \right) \\ & \\ \end{align}\]

\[\begin{align} & \text{Let }f\left( x \right)=\left\{ \begin{matrix} -{{x}^{3}}+1,\text{ }x\le 0 \\ -{{x}^{2}}+2x,\text{ }x>0 \\ \end{matrix} \right. \\ & \left( \text{a} \right) \\ & \text{Differentiate } \\ & f'\left( x \right)=\left\{ \begin{matrix} \frac{d}{dx}\left[ -{{x}^{3}}+1 \right],\text{ }x\le 0 \\ \frac{d}{dx}\left[ -{{x}^{2}}+2x \right],\text{ }x>0 \\ \end{matrix} \right. \\ & f'\left( x \right)=\left\{ \begin{matrix} -3{{x}^{2}},\text{ }x\le 0 \\ -2x+2,\text{ }x>0 \\ \end{matrix} \right. \\ & \text{Tthe critical points are: }x=0\text{ and}-2x+2=0 \\ & -2x+2=0 \\ & x=1 \\ & \text{We obtain the critical points }x=0,\text{ and }x=1 \\ & \text{Set the intervals }\left( -\infty ,0 \right),\left( 0,1 \right)\text{ and }\left( 1,\infty \right) \\ & \\ & \left( \text{b} \right) \\ & \text{Making a table of values }\left( \text{See examples on page 180 } \right) \\ & \begin{matrix} \text{Interval} & \left( -\infty ,0 \right) & \left( 0,1 \right) & \left( 1,\infty \right) \\ \text{Test Value} & x=-1 & x=0.5 & x=2 \\ \text{Sign of }f'\left( x \right) & \text{ }f'\left( -1 \right)<0 & \text{ }f'\left( 0.5 \right)>0 & \text{ }f'\left( 2 \right)<0 \\ \text{Conclusion} & \text{Decreasing} & \text{Increasing} & \text{Decreasing} \\ \end{matrix} \\ & \\ & \text{Therefore} \\ & f\left( x \right)\text{has a relative minimum at }\left( 0,f\left( 0 \right) \right) \\ & f\left( 0 \right)=-{{\left( 0 \right)}^{3}}+1=1\to \left( 0,1 \right) \\ & f\left( x \right)\text{has a relative maximum at }\left( 1,f\left( 1 \right) \right) \\ & f\left( 1 \right)=-{{\left( 1 \right)}^{2}}+2\left( 1 \right)=1\to \left( 1,1 \right) \\ & \\ & \left( \text{a} \right)x=0,\text{ }x=1 \\ & \left( \text{b} \right)\text{Decreasing on: }\left( -\infty ,0 \right)\text{ and }\left( 1,\infty \right)\text{ } \\ & \text{Increasing on: }\left( 0,1 \right),\text{ } \\ & \left( \text{c} \right)\text{relative maximum}\left( 1,1 \right) \\ & \text{ relative minimum}\left( 0,1 \right) \\ \end{align}\]