Explanation Formula used: Lagrange Multipliers: To obtain the relative extremum of the function f ( x , y , z ) subject to the constraint g ( x , y , z ) = 0 (assuming that these extreme values exist), (1) Form an auxiliary function F ( x , y , z , λ ) = f ( x , y , z ) + λ g ( x , y , z ) 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 , F z = 0 and F λ = 0 for all values of x , y , z and λ . (3) The solutions found in step (2) are candidates for the extremum of f . Calculation: Consider the function g ( x , y , z ) = 2 x y z + 12 y + z − 84 . Given that 2 x y z + 12 y + z − 84 . Obtain the value of y explicitly in terms of x from the function is g ( x , y ) = 0 . The auxiliary function is, F ( x , y , z , λ ) = f ( x , y , z ) + λ g ( x , y , z ) = x y z + λ ( 2 x y z + 12 y + z − 84 ) The above expression is called the Lagrange function while λ be the Lagrange multipliers. Compute the term F x from the Lagrangian function as given below. Differentiate F ( x , y , z , λ ) = x y z + λ ( 2 x y z + 12 y + z − 84 ) with respect to x , F x = ∂ ∂ x ( x y z + λ ( 2 x y z + 12 y + z − 84 ) ) = ∂ ∂ x ( x y z ) + ∂ ∂ x ( λ ( 2 x y z + 12 y + z − 84 ) ) = y z + λ ( 2 y z ) = y z + 2 λ y z Compute the term F y from the Lagrangian function. Differentiate F ( x , y , z , λ ) = x y z + λ ( 2 x y z + 12 y + z − 84 ) with respect to y , F y = ∂ ∂ y ( x y z + λ ( 2 x y z + 12 y + z − 84 ) ) = ∂ ∂ y ( x y z ) + ∂ ∂ y ( λ ( 2 x y z + 12 y + z − 84 ) ) = x z + λ ( 2 x z + 12 ) = x z + 2 λ ( x z + 6 ) Compute the term F z from the Lagrangian function. Differentiate F ( x , y , z , λ ) = x y z + λ ( 2 x y z + 12 y + z − 84 ) with respect to z , F z = ∂ ∂ z ( x y z + λ ( 2 x y z + 12 y + z − 84 ) ) = ∂ ∂ z ( x y z ) + ∂ ∂ z ( λ ( 2 x y z + 12 y + z − 84 ) ) = x y + λ ( 2 x y + 1 ) Compute the term F λ from the Lagrangian function

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 14E

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To maximize: The function f(x,y,z)=xyz subject to the constraint of function g(x,y,z)=2xyz+12y+z−84 .

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Chapter 8.5, Problem 13E

Chapter 8.5, Problem 15E

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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 , z ) = x y z subject to the constraint of function g ( x , y , z ) = 2 x y z + 12 y + z − 84 .

Solution Summary: The author explains how to maximize the function f(x,y,z)=xyz, which does not maximize to the constraint of the Lagrange multiplier.

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To maximize: The function f(x,y,z)=xyz subject to the constraint of function g(x,y,z)=2xyz+12y+z−84 .

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Chapter 8 Solutions