Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 3 - Differentiation - 3.2 Exercises: 20

Answer

Rolle's Theorem can be applied; $c$ belongs to $\{-\frac{\pi}{2}, 0, \frac{\pi}{2}\}.$

Work Step by Step

$f(x)$ is continuous for all values of $x$ and is differentiable at every value of $x.$ $f(-\pi)=f(\pi)=1.$ Since $f(x)$ is continuous over $[-\pi , \pi]$ and differentiable over $(-\pi, \pi)$, applying Rolle's Theorem over the interval $[-\pi, \pi]$ guarantees the existence of at least one value $c$ such that $-\pi\lt c\lt \pi$ and $f'(c)=0.$ $f'(x)=-2\sin{2x}.$ $f'(x)=0\to -2\sin{2x}=0\to 2x=\pi k\to x=\frac{k}{2}\pi$ where k is any integer. Substituting values for $k$ and checking the specified interval gives us the following solution set: $\{-\frac{\pi}{2}, 0, \frac{\pi}{2}\}.$
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