Explanation Given information: The objective function f ( x , y ) = 3 x 2 + 2 x y − y 2 , subject to the constraint g ( x , y ) = 5 − 2 x − y = 0 Explanation: Use Lagrange multipliers method: Replace f ( x , y ) by an auxiliary function of three variables F ( x , y , λ ) as, F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) Substitute f ( x , y ) = 3 x 2 + 2 x y − y 2 and g ( x , y ) = 5 − 2 x − y = 0 in F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) . F ( x , y , λ ) = 3 x 2 + 2 x y − y 2 + λ ( 5 − 2 x − y ) Find the partial derivatives of F ( x , y , λ ) = 3 x 2 + 2 x y − y 2 + λ ( 5 − 2 x − y ) and set it equal to zero. The partial derivatives of F ( x , y , λ ) = 3 x 2 + 2 x y − y 2 + λ ( 5 − 2 x − y ) with respect to x by treating y and λ as constant, ∂ F ∂ x = 6 x + 2 y − 2 λ = 0 … (1) The partial derivatives of F ( x , y , λ ) = 3 x 2 + 2 x y − y 2 + λ ( 5 − 2 x − y ) with respect to y by treating x and λ as constant, ∂ F ∂ y = 2 x − 2 y − λ = 0

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ISBN: 9780137590810

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar, William Edward Tavernetti

Publisher: PEARSON

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Chapter 7, Problem 25RE

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The values of x and y that maximize the objective function 3x2+2xy−y2, subject to the constraint 5−2x−y=0 by using the method of Lagrange multipliers.

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Chapter 7, Problem 24RE

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The values of x and y that maximize the objective function 3 x 2 + 2 x y − y 2 , subject to the constraint 5 − 2 x − y = 0 by using the method of Lagrange multipliers.

Solution Summary: The author explains the Lagrange multipliers method of determining the values of x and y that maximize the objective function 3x2+2xy-y

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The values of x and y that maximize the objective function 3x2+2xy−y2, subject to the constraint 5−2x−y=0 by using the method of Lagrange multipliers.

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Chapter 7 Solutions