Calculus 10th Edition

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

Chapter 3 - Applications of Differentiation - 3.2 Exercises: 45

Answer

The Mean Value Theorem can be applied; $c=\dfrac{\pi}{2}.$

Work Step by Step

$f(x)$ is continuous for all values of $x$ and differentiable at every value of $x.$ $a=0; b=\pi\to f(a)=0$ and $f(b)=0.$ Since $f(x)$ is continuous over $[a, b]$ and differentiable over $(a, b)$, applying Mean Value Theorem over the interval $[a , b]$ guarantees the existence of at least one value c such that $a\lt c\lt b$ and $f′(c)=\dfrac{f(b)-f(a)}{b-a}.$ $f'(c)=\dfrac{0-0}{\pi-0}=0.$ $f'(x)=\cos{x}\to f'(c)=\cos{c}=0\to c=\dfrac{\pi}{2}.$
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