Chapter 3 - The Derivative In Graphing And Applications - 3.1 Analysis Of Functions I: Increase, Decrease, and Concavity - Exercises Set 3.1 - Page 195: 23

\[\begin{align} & \left( \mathbf{a} \right)\text{increasing on the interval }\left[ -\frac{1}{2},\infty \right) \\ & \left( \mathbf{b} \right)\text{decreasing on the interval }\left( -\infty ,-\frac{1}{2} \right]\text{ } \\ & \left( \mathbf{c} \right)\text{concave upward on the interval }\left( -2,1 \right) \\ & \left( \mathbf{d} \right)\text{concave downward on the intervals }\left( -\infty ,-2 \right), \left( 1,+\infty \right) \\ & \left( \mathbf{e} \right)\text{inflection points at }x=-2,\text{ }x=1 \\ \end{align}\]

\[\begin{align} & f\left( x \right)=\sqrt[3]{{{x}^{2}}+x+1} \\ & \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[ \sqrt[3]{{{x}^{2}}+x+1} \right] \\ & f'\left( x \right)=\frac{1}{3}{{\left( {{x}^{2}}+x+1 \right)}^{-2/3}}\left( 2x+1 \right) \\ & f'\left( x \right)=\frac{2x+1}{3{{\left( {{x}^{2}}+x+1 \right)}^{2/3}}} \\ & \text{Find the critical points, set }f'\left( x \right)=0 \\ & f'\left( x \right)=0 \\ & 2x+1=0 \\ & x=-\frac{1}{2} \\ & \text{Interval analysis }\left( -\infty ,-\frac{1}{2} \right),\text{ }\left( -\frac{1}{2},\infty \right) \\ & f''\left( x \right)=\frac{d}{dx}\left[ \frac{2x+1}{3{{\left( {{x}^{2}}+x+1 \right)}^{2/3}}} \right] \\ & \text{Differentiate using wolfram alplha website} \\ & f''\left( x \right)=-\frac{2\left( {{x}^{2}}+x-2 \right)}{9{{\left( {{x}^{2}}+x+1 \right)}^{5/3}}} \\ & f''\left( x \right)=0 \\ & {{x}^{2}}+x-2=0 \\ & \text{By the quadratic formula} \\ & x=-2,\text{ }x=1 \\ & \text{We obtain the sign analysis shown in the following tables} \\ & \begin{matrix} \text{Interval} & \left( -\infty ,-\frac{1}{2} \right) & \left( -\frac{1}{2},\infty \right) \\ \text{Test Value} & x=-1 & x=0 \\ \text{Sign of }f'\left( x \right) & - & + \\ \text{Conclusion} & \text{Decreasing} & \text{Increasing} \\ \end{matrix} \\ & and \\ & \begin{matrix} \text{Interval} & \left( -\infty ,-2 \right) & \left( -2,1 \right) & \left( 1,+\infty \right) \\ \text{Test Value} & -5 & 0 & 5 \\ \text{Sign of }f''\left( x \right) & - & + & - \\ \text{Conclusion} & \text{C}\text{. downward} & \text{C}\text{. upward} & \text{C}\text{. downward} \\ \end{matrix} \\ & \\ & \text{Summary:} \\ & \left( \mathbf{a} \right)\text{ }f\left( x \right)\text{ is increasing on the interval }\left[ -\frac{1}{2},\infty \right) \\ & \left( \mathbf{b} \right)\text{ }f\left( x \right)\text{ is decreasing on the intervals }\left( -\infty ,-\frac{1}{2} \right]\text{ } \\ & \left( \mathbf{c} \right)\text{ }f\left( x \right)\text{ is concave upward on the interval }\left( -2,1 \right)\\ & \left( \mathbf{d} \right)\text{ }f\left( x \right)\text{ is concave downward on the intervals }\left( -\infty ,-2 \right), \left( 1,+\infty \right) \\ & \left( \mathbf{e} \right)\text{ Inflection points at }x=-2,\text{ }x=1 \\ \end{align}\]