   Chapter 14.8, Problem 15E

Chapter
Section
Textbook Problem

The method of Lagrange multipliers assumes that the extreme values exist, but that is not always the case. Show that the problem of finding the minimum value of f(x, y) = x2 + y2 subject to the constraint xy = 1 can be solved using Lagrange multipliers, but f docs not have a maximum value with that constraint.

To determine

To find: The minimum value of the function f(x,y)=x2+y2 subject to the constraint xy=1 and also show that there is no maximum value exist for the given constraint.

Explanation

Given:

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

Calculation:

The given function is f(x,y)=x2+y2 and g(x,y)=xy1 .

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

f(x,y)=λg(x,y)fx,fy=λgx,gyfx(x2+y2),fy(x2+y2)=λgx(xy1),gy(xy1)2x,2y=λy,x

Thus, the value of f(x,y)=λg(x,y) is 2x,2y=λy,x

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