Langrange Multiplier Method

Langrange Multiplier Method - The Lagrange Multiplier...

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The Lagrange Multiplier Method © Arizona State University, Department of Mathematics and Statistics 1 Objective: At the end of this lesson, you should be able to find the absolute maximum and minimum using the direct substitution method and the Lagrangian method. Background So far, we have optimized a function subject to some constraint using the direct substitution method. This works well as long as the constraint can be solved for one of the variables involved. Example 1. Find the maximum and minimum values of ( ) , 45 f x y xy = under the constraint 22 2 3 5 10 x xy y ++= It would be difficult to solve this problem using the direct substitution method. However, the Lagrange method is much easier. This will be shown later in this lecture. 2. Find the minimum value of ( ) , 34 f xy x y = + under the constraint 4 5 20 xy += . Now this constraint can be easily solved for either x or y and the direct substitution method works fine. Solve the constraint 4 5 20 to get 4 4 5 x y = . Then substitute. The result is to the right. Then set ( ) 278 128 '0 25 5 fx x = −= . Solve to get 2.302 x = , Back substitute to find ( ) 4 2.302 4 2.158 5 y = = . Evaluate to find the minimum value: ( ) ( ) ( ) 2.302,2.158 3 2.302 4 2.158 34.525 f =+= . State that the minimum value of the function is 34.525 at the point ( ) 2.302,2.158 . Finish the problem! The Lagrange Theorem Suppose you are given the function f ( x , y ) subject to the constraining function g ( x , y ) = c . Suppose both functions have continuous partial derivatives on a domain A in the xy -plane. Suppose that the point x 0 , y 0 ( ) is an interior extreme point the domain under the constraint g ( x , y ) = c and g x x 0 , y 0 ( ) 0 and g y x 0 , y 0 ( ) 0 (one them could be zero). Then there is a unique constant λ (lambda to
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This note was uploaded on 06/16/2011 for the course MAT 211 taught by Professor Seal during the Summer '08 term at ASU.

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Langrange Multiplier Method - The Lagrange Multiplier...

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