Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 4 - Applications of the Derivative - 4.3 The Mean Value Theorem and Monotonicity - Exercises - Page 189: 23


Critical point $c=3$ is a point of local maximum.

Work Step by Step

$f'(x)=6-2x$ is defined everywhere. $f'(x)=0\implies x=3$ is the critical point of $f(x)$. $f'(2.99)=6-(2\times2.99)=positive$ $f'(3.01)=6-(2\times3.01)=negative$ The sign change of $f'(x)$ at the critical point is positive to negative. Therefore, the critical point is a point of local maximum.
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