Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.7 Maximum and Minimum Values - 14.7 Exercises - Page 1008: 26


Maximum value: $f(\dfrac{\pi}{6},\dfrac{\pi}{6})=\dfrac{3}{2}$

Work Step by Step

Second derivative test: Some noteworthy points to calculate the local minimum, local maximum and saddle point of $f$. 1. If $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \gt 0$ and $f_{xx}(p,q)\gt 0$ , then $f(p,q)$ is a local minimum. 2.If $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \gt 0$ and $f_{xx}(p,q)\lt 0$ , then $f(p,q)$ is a local maximum. 3. If $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \lt 0$ , then $f(p,q)$ is a not a local minimum and local maximum or, a saddle point. For $(x,y)=(\dfrac{\pi}{6},\dfrac{\pi}{6})$ $D=\dfrac{3}{4}\gt 0$ ; and $f_{xx} =-1\lt 0$ Thus, when $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \gt 0$ and $f_{xx}(p,q)\lt 0$ , then $f(p,q)$ is a local maximum. Also, $f(x,y)=sin x+sin y +cos(x+y)=sin x+ siny+cosx \cos y-\sin x \sin y$ This yields, $f(\dfrac{\pi}{6},\dfrac{\pi}{6})=\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{\sqrt 3}{2}\dfrac{\sqrt 3}{2}-\dfrac{1}{2}\dfrac{1}{2}=\dfrac{3}{2}$ Therefore, we have Maximum value: $f(\dfrac{\pi}{6},\dfrac{\pi}{6})=\dfrac{3}{2}$
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