Explanation Given: The function f ( x , y ) = 6 x y subject to y − x = 6 . Result used: Method of Lagrange Multipliers: 1. Rewrite the problem in the form of maximize (or minimize) the function z = f ( x , y ) subject to the constraint g ( x , y ) = 0 . 2. Form the function F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) . 3. Compute the critical points of F ; That is, the solution of the system of the equation F x ( x , y , z ) = 0 F y ( x , y , z ) = 0 F z ( x , y , z ) = 0 4. if ( x 0 , y 0 , λ 0 ) is the only critical point of the function F , then the point ( x 0 , y 0 ) is the solution of the given hypothesis. If F has more than one critical point, compute the value of f ( x , y ) at ( x 0 , y 0 ) for each critical point ( x 0 , y 0 , λ 0 ) of F . then assume that the largest value of these values is the maximum value of f ( x , y ) , subject to constraint g ( x , y ) = 0 , and the smallest is the minimum of f ( x , y ) , subject to the constraint g ( x , y ) = 0 . Calculation: Rewrite the constraint as g ( x , y ) = 0 . y − x = 6 y − x − 6 = 0 The function f ( x , y ) = 6 x y subject to the constraint y − x − 6 = 0 . The form of the function should be F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) . Substitute f ( x , y ) = 6 x y and g ( x , y ) = y − x − 6 in F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) , F ( x , y , λ ) = 6 x y + λ ( y − x − 6 ) = 6 x y + λ y − λ x − λ 6 Thus, the function F ( x , y , λ ) = 6 x y + λ y − λ x − λ 6 . The partial derivative of the function F ( x , y , λ ) with respect to x is, F x ( x , y , λ ) = ∂ ∂ x ( 6 x y + λ y − λ x − λ 6 ) = ∂ ∂ x ( 6 x y ) + ∂ ∂ x ( λ y ) − ∂ ∂ x ( λ x ) − ∂ ∂ x ( λ 6 ) = ( 6 y ) ∂ ∂ x ( x ) + ∂ ∂ x ( λ y ) − ( λ ) ∂ ∂ x ( x ) − ∂ ∂ x ( λ 6 ) = 6 y − λ Thus, the partial derivative of the function F x ( x , y , λ ) = 6 y − λ

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 7.4, Problem 8E

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Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 7.1 - Prob. 1MP Ch. 7.1 - Matched Problem 2 Find f(2, 2, 3) for f in Example...Ch. 7.1 - Prob. 3MP Ch. 7.1 - Matched Problem 4 For the box in Example 4, find...Ch. 7.1 - Matched Problem 5 Refer to Example 5. Find the...Ch. 7.1 - Matched Problem 6 Find the coordinates of the...Ch. 7.1 - Matched Problem 7 (A) Describe the cross sections...Ch. 7.1 - Explore and Discuss 1 Imagine that you are facing...Ch. 7.1 - In Problems 18, find the indicated value of the...Ch. 7.1 - Prob. 2E

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Solution Summary: The author explains the Lagrange multipliers' method of finding the minimum of the function subject to the given constraints.

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