Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 14 - Partial Derivatives - 14.8 Exercises - Page 988: 33

Answer

$(\dfrac{100}{3} ,\dfrac{100}{3} ,\dfrac{100}{3})$

Work Step by Step

Use Lagrange Multipliers Method: $\nabla f=\lambda \nabla g$ 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 not a local minimum, local maximum, or a saddle point. From the given problem $z=100-x-y$ $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \gt 0$ and $f_{xx}(p,q)\lt 0$ , then $f(p,q)$ is a local maximum. $D(\dfrac{100}{3},\dfrac{100}{3})=\dfrac{200}{3} \gt 0$ and $f_{xx}=-\dfrac{200}{3} \lt 0$ The required points are: $(\dfrac{100}{3} ,\dfrac{100}{3} ,\dfrac{100}{3})$
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