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 ) = 2 x + 3 y − 6 . Given that 2 x + 3 y − 6 = 0 . hence, 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 , λ ) = f ( x , y ) + λ g ( x , y ) = x y + λ ( 2 x + 3 y − 6 ) 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 , λ ) = x y + λ ( 2 x + 3 y − 6 ) with respect to x , F x = ∂ ∂ x ( x y + λ ( 2 x + 3 y − 6 ) ) = ∂ ∂ x ( x y ) + ∂ ∂ x ( λ ( 2 x + 3 y − 6 ) ) = y + λ ( 2 ) = y + 2 λ By the step (2) of the above formula it is observed that y + λ 2 = 0 . Compute the term F y from the Lagrangian function. Differentiate F ( x , y , λ ) = x y + λ ( 2 x + 3 y − 6 ) with respect to y , F y = ∂ ∂ y ( x y + λ ( 2 x + 3 y − 6 ) ) = ∂ ∂ y ( x y ) + ∂ ∂ y ( λ ( 2 x + 3 y − 6 ) ) = x + λ ( 3 ) = x + 3 λ By the step (2) of the above formula it is observed that x + 3 λ = 0

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

10th Edition

ISBN: 9781285464640

Author: Soo T. Tan

Publisher: Brooks Cole

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Chapter 8.5, Problem 8E

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To maximize: The function f(x,y)=xy subjects to the constraint 2x+3y−6=0 .

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Chapter 8.5, Problem 7E

Chapter 8.5, Problem 9E

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Chapter 8 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Ch. 8.1 - What is a function of two variables? Give an...Ch. 8.1 - If f is a function of two variables, what can you...Ch. 8.1 - Prob. 3CQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Let g(x, y) = 2x2 y2. Compute g(1, 2), g(2,1),...Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Let h(s, t) = s ln t t ln s. Compute h(1, e),...

Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 18E Ch. 8.1 - In Exercises 19-24, sketch the level curves of the...Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - In Exercises 19-24, sketch the level curves of the...Ch. 8.1 - Prob. 24E Ch. 8.1 - Find an equation of the level curve of f (x, y) =...Ch. 8.1 - Prob. 26E Ch. 8.1 - In Exercises 27 and 28, match the graph of the...Ch. 8.1 - Prob. 28E Ch. 8.1 - Can two level curves of a function f of two...Ch. 8.1 - Prob. 30E Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - PRICE-TO-EARNINGS Ratio The current PE ratio...Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - Prob. 36E Ch. 8.1 - COST FUNCTION FOR A LOUDSPEAKER SYSTEM Acrosonic...Ch. 8.1 - REVENUE FUNCTION FOR ELECTRIC CARS Bell Motors...Ch. 8.1 - 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Prob. 41E Ch. 8.5 - Prob. 42E Ch. 8.6 - Prob. 1CQ Ch. 8.6 - Prob. 2CQ Ch. 8.6 - Prob. 3CQ Ch. 8.6 - Prob. 4CQ Ch. 8.6 - Prob. 5CQ Ch. 8.6 - Prob. 1E Ch. 8.6 - Prob. 2E Ch. 8.6 - Prob. 3E Ch. 8.6 - Prob. 4E Ch. 8.6 - Prob. 5E Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - Prob. 7E Ch. 8.6 - Prob. 8E Ch. 8.6 - Prob. 9E Ch. 8.6 - Prob. 10E Ch. 8.6 - Prob. 11E Ch. 8.6 - Prob. 12E Ch. 8.6 - Prob. 13E Ch. 8.6 - Prob. 14E Ch. 8.6 - Prob. 15E Ch. 8.6 - Prob. 16E Ch. 8.6 - Prob. 17E Ch. 8.6 - Prob. 18E Ch. 8.6 - Prob. 19E Ch. 8.6 - Prob. 20E Ch. 8.6 - Prob. 21E Ch. 8.6 - Prob. 22E Ch. 8.6 - Prob. 23E Ch. 8.6 - Prob. 24E Ch. 8.6 - Prob. 25E Ch. 8.6 - Prob. 26E Ch. 8.6 - Prob. 27E Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - Prob. 30E Ch. 8.6 - Prob. 31E Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - Prob. 33E Ch. 8.6 - Prob. 34E Ch. 8.6 - Prob. 35E Ch. 8.6 - Prob. 36E Ch. 8.6 - 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To maximize: The function f ( x , y ) = x y subjects to the constraint 2 x + 3 y − 6 = 0 .

Solution Summary: The author explains the Lagrange Multipliers to obtain the relative extremum of the function f(x,y) subjected to the constraint 2x+3y-6=0

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To maximize: The function f(x,y)=xy subjects to the constraint 2x+3y−6=0 .

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Chapter 8 Solutions