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: 5


$ c= ±\sqrt {7}$

Work Step by Step

The slope of the secant line is $\frac{f(b)-f(a)}{b-a}=\frac{f(5)-f(-4)}{5-(-4)}=\frac{5^{3}-(-4)^{3}}{9}=21$ We must find $ c $ such that $ f'(c)=21$. $ f'(c)=3c^{2}=21\implies c^{2}=\frac{21}{3}=7$ $\implies c= ±\sqrt {7}$ The value $ c=±\sqrt {7}$ lies in (-4, 5) and satisfies $ f'(±\sqrt {7})=21$. This satisfies the conclusion of the MVT.
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