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: 18

Answer

Rolle's Theorem can be applied; $c=\pi$.

Work Step by Step

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