Explanation Given: Thefunction to minimize f ( x , y ) = x 2 − x y + 2 y 2 subject to 2 x + y = 22 . Concept Used: To optimize (that is, to find the minimum and maximum value of) a function f ( x , y ) subject to the constraint g ( x , y ) = k Solve the following system of equations ∇ f ( x , y ) = λ ∇ g ( x , y ) g ( x , y ) = k where λ is the Lagrange Multiplier. Plug all the solutions ( x , y ) from the above step into f ( x , y ) . The maximum value of the function f ( x , y ) subject to the constraint g ( x , y ) = k is the maximum value of f ( x , y ) after plugging all the solutions ( x , y ) into f ( x , y ) . Calculation: For the givenfunction f ( x , y ) = x 2 − x y + 2 y 2 Calculating the first partial derivatives f x and f y of f ( x , y ) f ( x , y ) = x 2 − x y + 2 y 2 that is, f x ( x , y ) = 2 x − y and, f y ( x , y ) = 4 y − x and, calculating the first partial derivatives g x and g y of g ( x , y ) g ( x , y ) = 2 x + y that is, g x ( x , y ) = 2 and, g y ( x , y ) = 1 Substituting the values in the equation f x ( x , y ) = λ g x ( x , y ) that is, 2 x − y = λ ( 2 ) y = 2 x − 2 λ - - - - - ( 1 ) and, the values in the equation f y ( x , y ) = λ g y ( x , y ) that is, 4 y − x = λ ( 1 ) y = x + λ 4 �

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ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 11.8, Problem 10PS

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To find:Minimize f(x,y)=x2−xy+2y2 subject to constraint 2x+y=22 using the method of Lagrange multipliers.

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Chapter 11.8, Problem 9PS

Chapter 11.8, Problem 11PS

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Chapter 11 Solutions

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Ch. 11.1 - Prob. 1PS Ch. 11.1 - Prob. 2PS Ch. 11.1 - Prob. 3PS Ch. 11.1 - Prob. 4PS Ch. 11.1 - Prob. 5PS Ch. 11.1 - Prob. 6PS Ch. 11.1 - Prob. 7PS Ch. 11.1 - Prob. 8PS Ch. 11.1 - Prob. 9PS Ch. 11.1 - Prob. 10PS

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Minimize f ( x , y ) = x 2 − x y + 2 y 2 subject to constraint 2 x + y = 22 using the method of Lagrange multipliers.

Solution Summary: The author explains how to minimize f(x,y) using the method of Lagrange multipliers.

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To find:Minimize f(x,y)=x2−xy+2y2 subject to constraint 2x+y=22 using the method of Lagrange multipliers.

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Chapter 11 Solutions