Explanation Given: Thefunction to maximize f ( x , y ) = x 2 + y 2 subject to x + y − 1 = 0 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 ( x , y ) = λ g x ( x , y ) that is,

Student's Solution and Survival Manual for Calculus

7th Edition

ISBN: 9781524934040

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

Publisher: Kendall Hunt Publishing

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Chapter 11.8, Problem 8PS

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Tofind: Maximize f(x,y)=x2+y2 subject to constraint x+y−1=0 using the method of Lagrange multipliers.

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Chapter 11.8, Problem 7PS

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Chapter 11 Solutions

Student's Solution and Survival Manual for Calculus

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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Tofind: Maximize f ( x , y ) = x 2 + y 2 subject to constraint x + y − 1 = 0 using the method of Lagrange multipliers.

Solution Summary: The author explains the Lagrange multipliers' method of maximizing f(x,y) to find the minimum and maximum value of a function.

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Tofind: Maximize f(x,y)=x2+y2 subject to constraint x+y−1=0 using the method of Lagrange multipliers.

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Chapter 11 Solutions