Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 4 - Applications of the Derivative - 4.2 Extreme Values - Exercises - Page 181: 15



Work Step by Step

Let $\sin \theta =u$ Then, $g(\theta)=\sin^{2}\theta=u^{2}$ $g'(\theta)=\frac{d}{d\theta}(u^{2})=2u\frac{du}{d\theta}=2\sin\theta\frac{d}{d\theta}(\sin\theta)$ $=2\sin\theta \cos\theta$ If $g'(\theta)$ does not exist or $g'(\theta)=0$, this implies $\theta$ is a critical point of the function. But, $g'(\theta)$ exists for all $\theta$. If $g'(\theta)=2 \sin\theta\cos \theta=0$ $\implies \sin\theta=0$ or $\cos \theta= 0$ $\implies \theta =n\pi$ or $\theta =(2n+1)\frac{\pi}{2}$ where $n$ is any integer. Combining $\theta=n\pi$ and $\theta=(2n+1)\frac{\pi}{2}$, we can write $g'(\theta)=0\implies \theta=\frac{n\pi}{2}$
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