Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 3 - The Derivative In Graphing And Applications - 3.1 Analysis Of Functions I: Increase, Decrease, and Concavity - Exercises Set 3.1 - Page 195: 19

Answer

\[\begin{align} & \left( \mathbf{a} \right)\text{increasing on the interval }\left[ 1,+\infty \right) \\ & \left( \mathbf{b} \right)\text{decreasing on the interval }\left( -\infty ,1 \right] \\ & \left( \mathbf{c} \right)\text{concave upward on the interval }\left( -\infty ,0 \right),\left( \frac{2}{3},+\infty \right) \\ & \left( \mathbf{d} \right)\text{concave downward on the interval }\left( 0,\frac{2}{3} \right) \\ & \left( \mathbf{e} \right)\text{inflection points at }x=0,\text{ }x=\frac{2}{3} \\ \end{align}\]

Work Step by Step

\[\begin{align} & f\left( x \right)=3{{x}^{4}}-4{{x}^{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[ 3{{x}^{4}}-4{{x}^{3}} \right] \\ & f'\left( x \right)=12{{x}^{3}}-12{{x}^{2}} \\ & \text{Find the critical points, set }f'\left( x \right)=0 \\ & f'\left( x \right)=0 \\ & 12{{x}^{3}}-12{{x}^{2}}=0 \\ & 12{{x}^{2}}\left( x-1 \right)=0 \\ & x=0,\text{ }x=1 \\ & \text{Interval analysis }\left( -\infty ,0 \right),\text{ }\left( 0,1 \right),\text{ }\left( 1,\infty \right) \\ & f''\left( x \right)=\frac{d}{dx}\left[ 12{{x}^{3}}-12{{x}^{2}} \right] \\ & f''\left( x \right)=36{{x}^{2}}-24x \\ & f''\left( x \right)=0 \\ & 36{{x}^{2}}-24x=0 \\ & 12x\left( 3x-2 \right)=0 \\ & x=0,\text{ }x=\frac{2}{3} \\ & \text{We obtain the sign analysis shown in the following tables} \\ & \\ & \begin{matrix} \text{Interval} & \left( -\infty ,0 \right) & \left( 0,1 \right) & \left( 1,+\infty \right) \\ \text{Test Value} & -1 & 1/2 & 2 \\ \text{Sign of }f'\left( x \right) & - & - & + \\ \text{Conclusion} & \text{Decreasing} & \text{Decreasing} & \text{Increasing} \\ \end{matrix} \\ & \\ & \begin{matrix} \text{Interval} & \left( -\infty ,0 \right) & \left( 0,\frac{2}{3} \right) & \left( \frac{2}{3},+\infty \right) \\ \text{Test Value} & -1 & 1/2 & 1 \\ \text{Sign of }f''\left( x \right) & + & - & + \\ \text{Conclusion} & \text{C}\text{. upward} & \text{C}\text{. downward} & \text{C}\text{. upward} \\ \end{matrix} \\ & \\ & \text{Summary:} \\ & \left( \mathbf{a} \right)\text{ }f\left( x \right)\text{ is increasing on the interval }\left[ 1,+\infty \right) \\ & \left( \mathbf{b} \right)\text{ }f\left( x \right)\text{ is decreasing on the interval }\left( -\infty ,1 \right] \\ & \left( \mathbf{c} \right)\text{ }f\left( x \right)\text{ is concave upward on the interval }\left( -\infty ,0 \right),\left( \frac{2}{3},+\infty \right) \\ & \left( \mathbf{d} \right)\text{ }f\left( x \right)\text{ is concave downward on the interval }\left( 0,\frac{2}{3} \right) \\ & \left( \mathbf{e} \right)\text{ Inflection points at }x=0,\text{ }x=\frac{2}{3} \\ \end{align}\]
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