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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s).

60. f ( x , y ) = 1 x + 1 y ; 1 x 2 + 1 y 2 = 1

To determine

To find: The extreme values of the function f(x,y)=1x+1y subject to the constraint 1x2+1y2=1 by using Lagrange multipliers.

Explanation

Given:

The function f(x,y)=1x+1y which is subject to the constraint g(x,y)=1x2+1y2=1 .

Result used:

“The Lagrange multipliers defined as f(x,y,z)=λg(x,y,z) . This equation can be expressed as fx=λgx , fy=λgy , fz=λgz and g(x,y,z)=k ”.

Calculation:

The Lagrange multipliers f(x,y)=λg(x,y) is computed as follows,

f(x,y)=λg(x,y)fx,fy=λgx,gyfx(1x+1y),fy(1x+1y)=λgx(1x2+1y2),gy(1x2+1y2)1x2,1y2=λ2x3,2y3

Thus, the value of f(x,y)=λg(x,y) is 1x2,1y2=λ2x3,2y3

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