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Q. 22

Expert-verified

Calculus

Found in: Page 985

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

When you use the method of Lagrange multipliers to find the maximum and minimum of $f (x, y) = x + y$ subject to the constraint $x y = 1,$ you obtain two points. Is there a relative maximum at one of the points and a relative minimum at the other? Which is which?

The maximum value of the function $f (x, y) = x + y$ subject to constraint $x y = 1$ is $2$ and the minimum value is $- 2 .$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

Given function is $f (x, y) = x + y .$

Given constraint is $x y = 1 .$

Step 2. critical points of the function.

Gradients of function.

$\nabla f (x, y) = i + j \nabla g (x, y) = y i + x j$

Use the method of Lagrange multipliers.

role="math" localid="1649887252906" $\nabla f (x, y) = λ \nabla g (x, y) i + j = λ (y i + x j) i + j = λ y i + λ x j$

Compare terms.

role="math" localid="1649887083426" $λ y = 1 \Rightarrow y = \frac{1}{λ} λ x = 1 \Rightarrow x = \frac{1}{λ} so x = y$

substitute $x = y$ in constraint.

$x y = 1 y^{2} = 1 y = \pm 1 x = \pm 1$

so critical points are role="math" localid="1649887209373" $(- 1, - 1) & (1, 1) .$

Step 3. maximum and minimum of a function.

Find function value at $(- 1, - 1) .$

role="math" localid="1649887515378" $f (x, y) = x + y f (- 1, - 1) = (- 1) + (- 1) f (- 1, - 1) = - 2$

Find the function value at $(1, 1) .$

$f (x, y) = x + y f (1, 1) = (1) + (1) f (1, 1) = 2$

So the maximum value of the function is $2$ and the minimum value is $- 2 .$

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