Chapter 3 - The Derivative In Graphing And Applications - 3.1 Analysis Of Functions I: Increase, Decrease, and Concavity - Exercises Set 3.1 - Page 195: 16

\[\begin{align} & \left( \mathbf{a} \right)\text{increasing on the interval }\left( -\infty ,-2 \right) \\ & \left( \mathbf{b} \right)\text{decreasing on the interval }\left( -2,\infty \right) \\ & \left( \mathbf{c} \right)\text{ none} \\ & \left( \mathbf{d} \right)\text{concave downward on the interval }\left( -\infty ,\infty \right) \\ & \left( \mathbf{e} \right)\text{ no inflection point} \\ \end{align}\]

\[\begin{align} & f\left( x \right)=5-4x-{{x}^{2}} \\ & \text{The domain of the function is }\left( -\infty ,\infty \right) \\ & \text{Calculate the first and second derivatives} \\ & f'\left( x \right)=\frac{d}{dx}\left[ 5-4x-{{x}^{2}} \right] \\ & f'\left( x \right)=-4-2x \\ & \text{Find the critical points, set }f'\left( x \right)=0 \\ & f'\left( x \right)=0 \\ & -4-2x=0 \\ & x=-2,\text{ interval analysis }\left( -\infty ,-2 \right),\text{ }\left( -2,\infty \right) \\ & f''\left( x \right)=\frac{d}{dx}\left[ -4-2x \right] \\ & f''\left( x \right)=-2 \\ & \text{We obtain the sign analysis shown in the following tables} \\ & \begin{matrix} \text{Interval} & \left( -\infty ,-2 \right) & \left( -2,\infty \right) \\ \text{Test Value} & x=-3 & x=0 \\ \text{Sign of }f'\left( x \right) & + & - \\ \text{Conclusion} & \text{Increasing} & \text{Decreasing} \\ \end{matrix} \\ & \\ & \begin{matrix} \text{Interval} & \left( -\infty ,-2 \right) & \left( -2,\infty \right) \\ \text{Test Value} & x=-3 & x=0 \\ \text{Sign of }f''\left( x \right) & - & - \\ \text{Conclusion} & \text{Concave downward} & \text{Concave downward} \\ \end{matrix} \\ & \\ & \text{Summary:} \\ & \left( \mathbf{a} \right)\text{ }f\left( x \right)\text{ is increasing on the interval }\left( -\infty ,-2 \right) \\ & \left( \mathbf{b} \right)\text{ }f\left( x \right)\text{ is decreasing on the interval }\left( -2,\infty \right) \\ & \left( \mathbf{c} \right)\text{ None} \\ & \left( \mathbf{d} \right)\text{ }f\left( x \right)\text{ is Concave downward on the interval }\left( -\infty ,\infty \right) \\ & \left( \mathbf{e} \right)\text{ There is no change in concavity and hence no inflection} \\ & \text{point}\text{.} \\ \end{align}\]