Explanation Formula used: Lagrange Multipliers: To obtain the relative extremum of the function f ( x , y ) subject to the constraint g ( x , y ) = 0 (assuming that these extreme values exist), (1) Form an auxiliary function F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) called the Lagrangian function where the variable λ is called the Lagrange multiplier. (2) Solve the system that consists of the equations F x = 0 , F y = 0 and F λ = 0 for all values of x , y , and λ . (3) The solutions found in step (2) are candidates for the extremum of f . Calculation: Consider the function g ( x , y ) = 9 x 2 + y 2 − 9 Given that 9 x 2 + y 2 = 9 , hence, obtain the value of y explicitly in terms of x from the function is g ( x , y ) = 0 . F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) = x y 2 + λ ( 9 x 2 + y 2 − 9 ) The above expression is called the Lagrange function while λ be the Lagrange multipliers. Solve the system of equations of function F to find critical points as follows. Compute the term F x from the Lagrangian function as given below. F x = ∂ ∂ x ( x y 2 + λ ( 9 x 2 + y 2 − 9 ) ) = y 2 + 9 ( 2 x ) λ = y 2 + 18 x λ By the step (2) of the above formula it is observed that y 2 + 18 x λ = 0 . Compute the term F y from the Lagrangian function. F y = ∂ ∂ y ( x y 2 + λ ( 9 x 2 + y 2 − 9 ) ) = 2 x y + ( 2 y ) λ = 2 x y + 2 λ y By the step (2) of the above formula it is observed that 2 x y + 2 λ y = 0 . Compute the term F λ from the Lagrangian function. F λ = ∂ ∂ λ ( x y 2 + λ ( 9 x 2 + y 2 − 9 ) ) = 9 x 2 + y 2 − 9 By the step (2) of the above formula it is observed that 9 x 2 + y 2 − 9 = 0 . To find the critical point(s) of the function F, solve the system composed of the equation. Obtain a value of λ from the equation y 2 + 18 x λ = 0 . y 2 + 18 x λ = 0 18 x λ = − y 2 λ = − y 2 18 x Obtain a value of λ from the equation 2 x y + 2 λ y = 0 . 2 x y + 2 λ y = 0 2 λ y = − 2 x y λ = − 2 x y 2 y λ = − x Equate the value of λ to obtain the values of x and y

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 8.5, Problem 9E

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To maximize: The function f(x,y)=xy2 subjects to the constraint 9x2+y2=9 .

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Chapter 8.5, Problem 8E

Chapter 8.5, Problem 10E

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Chapter 8 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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To maximize: The function f ( x , y ) = x y 2 subjects to the constraint 9 x 2 + y 2 = 9 .

Solution Summary: The author explains how to maximize the function f(x,y) with the Lagrange multiplier.

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To maximize: The function f(x,y)=xy2 subjects to the constraint 9x2+y2=9 .

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Chapter 8 Solutions