Chapter 3 - The Derivative In Graphing And Applications - 3.1 Analysis Of Functions I: Increase, Decrease, and Concavity - Exercises Set 3.1 - Page 195: 17

\[\begin{align} & \left( \mathbf{a} \right)\text{increasing on the interval }\left( -\infty ,\infty \right) \\ & \left( \mathbf{b} \right)\text{none} \\ & \left( \mathbf{c} \right)\text{concave upward on the interval }\left( -\frac{1}{2},+\infty \right) \\ & \left( \mathbf{d} \right)\text{concave downward on the interval }\left( -\infty ,-\frac{1}{2} \right) \\ & \left( \mathbf{e} \right)\text{inflection point at }x=-\frac{1}{2} \\ \end{align}\]

\[\begin{align} & f\left( x \right)={{\left( 2x+1 \right)}^{3}} \\ & \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[ {{\left( 2x+1 \right)}^{3}} \right] \\ & f'\left( x \right)=3{{\left( 2x+1 \right)}^{2}}\left( 2 \right) \\ & f'\left( x \right)=6{{\left( 2x+1 \right)}^{2}} \\ & \text{Find the critical points, set }f'\left( x \right)=0 \\ & f'\left( x \right)=6{{\left( 2x+1 \right)}^{2}} \\ & 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[ 6{{\left( 2x+1 \right)}^{2}} \right] \\ & f''\left( x \right)=12\left( 2x+1 \right)\left( 2 \right) \\ & f''\left( x \right)=24\left( 2x+1 \right) \\ & f''\left( x \right)=0 \\ & x=-\frac{1}{2} \\ & \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{Increasing} & \text{Increasing} \\ \end{matrix} \\ & \\ & \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{Concave downward} & \text{Concave upward} \\ \end{matrix} \\ & \\ & \text{Summary:} \\ & \left( \mathbf{a} \right)\text{ }f\left( x \right)\text{ is increasing on the interval }\left( -\infty ,\infty \right) \\ & \left( \mathbf{b} \right)\text{None} \\ & \left( \mathbf{c} \right)\text{ }f\left( x \right)\text{ is concave upward on the interval }\left( -\frac{1}{2},+\infty \right) \\ & \left( \mathbf{d} \right)\text{ }f\left( x \right)\text{ is concave downward on the interval }\left( -\infty ,-\frac{1}{2} \right) \\ &\left( \mathbf{e} \right)\text{ Inflection point at }x=-\frac{1}{2} \\ \end{align}\]