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: 7



Work Step by Step

Given $$y=x \sin x, \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ Since $$f'(x)=x \cos x+ \sin x $$ Then by MVT, there exists a constant $c\in (-\pi/2,\pi/2)$ such that \begin{align*} f'(c)&= \frac{f(b)-f(a) }{b-a}\\ c \cos c+ \sin c &=\frac{ \pi/2-\pi/2}{\pi}\\ c \cos c+ \sin c &=0 \end{align*} Then $c=0 \in(-\pi/2,\pi/2)$
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