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: 19


Local minimum $f(0,1)=f(\pi,-1)=f(2 \pi,1)=-1$ ; Saddle points: $(\dfrac{\pi}{2},0),(\dfrac{3\pi}{2},0)$

Work Step by Step

Given: $f(x,y)=y^2-2y \cos x$; $-1 \leq x \leq l$ To solve this problem we will take the help of Second derivative test that suggests the following conditions to determine the local minimum, local maximum and saddle points of $f(x,y)$ or $f(x,y,z)$. i) 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. ii) 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. iii) 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. a)For $(x,y)=(0,0)$; we have $D(0,0)=4 \gt 0$ and $f_{xx}(0,0)=2 \gt 0$ b) For $(x,y)=(0,1)$, we have $D(0,1)=4 \gt 0$ and $f_{xx}=2 \gt 0$ c) For $(x,y)=(\pi,-1)$; we have $D(\pi,-1)=4 \gt 0$ and $f_{xx}=2 \gt 0$ d) For $(x,y)=(2\pi,1)$; we have $D(\pi,-1)=4 \gt 0$ and $f_{xx}=2 \gt 0$ e) For $(x,y)=(\dfrac{\pi}{2},0)$; we have $D(\dfrac{\pi}{2},0)=-2 \lt 0$ f) For $(x,y)=(\dfrac{3\pi}{2},0)$, we have $D(\dfrac{\pi}{2},0)=-2 \lt 0$ From the above conditions we conclude that Local minimum $f(0,1)=f(\pi,-1)=f(2 \pi,1)=-1$ ; Saddle points : $(\dfrac{\pi}{2},0),(\dfrac{3\pi}{2},0)$
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