Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 14: Partial Derivatives - Section 14.8 - Lagrange Multipliers - Exercises 14.8 - Page 852: 2


$(\pm \sqrt 5, \sqrt 5) $ and $(\pm \sqrt 5,-\sqrt 5)$

Work Step by Step

Consider $f(x,y) =xy$ $g(x,y)=x^2+y^2-10=0$ ...(1) Consider the gradient equation as follows: $\nabla =\lambda \nabla g$ Here, $y=2 \lambda x$ or, $y=\pm \dfrac{1}{2}$ Now, $x=2 \lambda y$ or, $x=\pm y$ From equation (1), we have $g(x,y)=x^2+y^2-10=0$ This implies that $x=\pm \sqrt 5$ and $y= \pm \sqrt 5$ Points: $(\pm \sqrt 5, \sqrt 5) $ and $(\pm \sqrt 5,-\sqrt 5)$
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