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Q. 22

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 xy=1 is 2 and the minimum value is -2.

See the step by step solution

Step by Step Solution

Step 1. Given information.    

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

Given constraint is xy=1.

Step 2. critical points of the function.

Gradients of function.


Use the method of Lagrange multipliers.

role="math" localid="1649887252906" f(x,y)=λg(x,y)i+j=λ(yi+xj)i+j=λyi+λxj

Compare terms.

role="math" localid="1649887083426" λy=1y=1λλx=1x=1λso x=y

substitute x=y in constraint.


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+yf(-1,-1)=(-1)+(-1)f(-1,-1)=-2

Find the function value at (1,1).


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

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