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 - 3.2 Analysis Of Functions II: Relative Extrema; Graphing Polynomials - Exercises Set 3.2 - Page 206: 29

Answer

Relative maximum: $\frac{4}{3}$

Work Step by Step

First derivative \[ f^{\prime}(x)=8-6 x \] Critical points (zeros of first derivative or point where first derivative does not exist) \[ x=\frac{4}{3} \] Using first derivative test, determine relative extrema $f^{\prime}(x)>0$ on the left and $f^{\prime}(x)<0$ on the right of $x=\frac{4}{3},$ so $x=\frac{4}{3}$ is a relative maximum. Second derivative. \[ -6=f^{\prime \prime}(x) \] $f^{\prime}\left(\frac{4}{3}\right)=0$ and $f^{\prime \prime}\left(\frac{4}{3}\right)<0,$ so $x=\frac{4}{3}$ is a relative maximum.
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