Explanation Given: Thefunction to minimize f ( x , y ) = x 2 − y 2 subject to x 2 + y 2 = 4 Concept Used: To optimize (i.e., 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 i.e., 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 2 + y 2 i.e., g x ( x , y ) = 2 x and, g y ( x , y ) = 2 y Substituting the values in the equation f x ( x , y ) = λ g x ( x , y ) i.e., 2 x = λ ( 2 x ) 2 x − λ ( 2 x ) = 0 ( 2 x ) ( 1 − λ ) = 0 i

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ISBN: 9781465208880

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

Publisher: Kendall Hunt Publishing

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Chapter 11.8, Problem 11PS

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Tominimize f(x,y)=x2−y2 subject to constraint x2+y2=4 using the method of Lagrange multipliers.

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Chapter 11.8, Problem 10PS

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Chapter 11 Solutions

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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Tominimize f ( x , y ) = x 2 − y 2 subject to constraint x 2 + y 2 = 4 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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Tominimize f(x,y)=x2−y2 subject to constraint x2+y2=4 using the method of Lagrange multipliers.

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