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 - 3.1 Exercises: 15


The function has three critical numbers in the specified domain: {$\frac{\pi}{3}, \pi, \frac{5\pi}{3} $}.

Work Step by Step

$h'(x)=2(\sin{x})(\cos{x})-\sin{x}=\sin{x}(2\cos{x}-1).$ $h'(x)=0\to \sin{x}(2\cos{x}-1)=0\to\sin{x}=0$ or $\cos{x}=\frac{1}{2}.$ $\sin{x}=0\to x=\pi k$ where $k$ is any integer. $\cos{x}=\frac{1}{2}\to x=\frac{\pi}{3}+2\pi k$ or $x=\frac{5\pi}{3}+2\pi k.$ Plugging in values for $k$ and remembering the limits on x in this problem, we get the following ciritical numbers for $h(x):$ {$\frac{\pi}{3}, \pi, \frac{5\pi}{3} $}
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