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: 13

Answer

False

Work Step by Step

\[\begin{align} & \text{The statement is false, because the derivative }f'\left( 1 \right)\text{ can be 0} \\ & \text{on the interval }\left[ 0,2 \right],\text{ for example consider }\\ &f\left( x \right)=\text{ 2}{{\left( x-1 \right)}^{5}} \\ & f'\left( x \right)=10{{\left( x-1 \right)}^{4}} \\ & \text{Critical point }x=1 \\ & f'\left( 0 \right)=+,\text{ }f'\left( 2 \right)=+.\text{ The function }f\left( x \right)\text{ is increasing on} \\ & \left[ 0,2 \right]\text{ but }f'\left( 1 \right)=10{{\left( 1-1 \right)}^{4}}=0 \\ & \text{False} \\ \end{align}\]
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.