Explanation Given: Thefunction to minimize f ( x , y ) = x 2 + y 2 subject to x + y = 2 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 + y 2 Calculating the first partial derivatives f x and f y of f ( x , y ) f ( x , y ) = x 2 + y 2 that is, f x ( x , y ) = 2 x and, f y ( x , y ) = 2 y and, calculating the first partial derivatives g x and g y of g ( x , y ) g ( x , y ) = x + y that is, g x ( x , y ) = 1 and, g y ( x , y ) = 1 Substituting the values in the equation f x

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Chapter 11.8, Problem 4PS

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Tofind: Minimize f(x,y)=x2+y2 subject to constraint x+y=2 using the method of Lagrange multipliers.

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Chapter 11.8, Problem 3PS

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Tofind: Minimize f ( x , y ) = x 2 + y 2 subject to constraint x + y = 2 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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Tofind: Minimize f(x,y)=x2+y2 subject to constraint x+y=2 using the method of Lagrange multipliers.

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