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: 33

Answer

Relative maximum: $x=1$ Relative minima: $x=2$ and $x=0$

Work Step by Step

First derivative \[ f^{\prime}(x)=-12 x^{2}+8 x+4 x^{3} \] Critical points (zeros of first derivative or point where first derivative does not exist) \[ x=2, x=1 \text { and } x=0 \] Second derivative \[ f^{\prime \prime}(x)=8-24 x+12 x^{2} \] Values of second derivative at critical points: $f^{\prime \prime}(0)=8>0$ \[ f^{\prime \prime}(1)=-24+8-4+12<0 \] $f^{\prime \prime}(2)=8>0$ \[ \begin{array}{l} f^{\prime \prime}(1)=-24+8-4+12<0 \\ f^{\prime \prime}(2)=8>0 \end{array} \] Second derivative test Relative minima: $x=2$ and $x=0$ Relative maximum: $x=1$
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