UNIT8 - Unit 8 Lagranges Method and Calculus of...

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Unit 8 Lagrange’s Method and Calculus of Trigonometric Functions Constrained Optimization: Lagrange’s Method Old Problem: Minimize z = x 2 + y 2 +2 subject to the constraint x + y =8. Last time we solved a similar problem by using the constraint to eliminate avar iab le . i.e. the problem becomes: Minimize: z = x 2 +(8 x ) 2 +2 Now easy to solve: x =4 y , z New Problem: Minimize f ( x, y, z )= xyz subject to the constraint xy xz +4 yz = 216. The old method does not work here since we can not use the constraint to solve for one of the variables, thus we require a new method. To this end we turn to a method developed by Joseph Lagrange in the 1700’s. Lagrange’s Method For the problem of optimizing z = f ( x, y ) subject to the constraint g ( x, y )=0,let 1
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F ( x, y, λ )= f ( x, y )+ λg ( x, y ) The local extreme points ( x, y ) will be found in the set of points ( x, y, λ ) that satisfy the three equations: F x ( x, y, λ )=0 F y ( x, y, λ F λ ( x, y, λ This process generalizes to the case where we have f ( x, y, z ). Example 1. Minimize f ( x, y x 2 + y 2 +2 subject to g ( x, y x + y 8=0. Using Lagrange’s Method, F ( x, y, λ ( x 2 + y 2 +2 ) + λ ( x + y 8) F x =2 x + λ =0 λ = 2 x (1) F y y + λ λ = 2 y (2) F λ = x + y 8=0 (3) Setting (1)=(2) we get 2 x = 2 y x = y Substituting this into (3) yields: x + x 8=0 x =4 And since we have x = y ,w eg e t y = 4 as well. So our solution is ( x, y )=(4 , 4). 2
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We may generalize our attack above and say that a good approach to these questions is to solve all but the last equation for λ andthenequa te .
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This note was uploaded on 04/15/2010 for the course MATH 20C taught by Professor Lit during the Spring '10 term at UCLA.

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UNIT8 - Unit 8 Lagranges Method and Calculus of...

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