Explanation Given: Function is f ( x , y ) = 2 x + 2 x y + y and Constraint is 2 x + y = 100 . Formula Used: Gradient vector of a function f is expressed as: ∇ f ( x , y ) = f x ( x , y ) i + f y ( x , y ) j Calculation: Consider the objective function be represented by z = f ( x , y ) and constraint equation by g ( x , y ) = c where c is a constant. Equations are: f ( x , y ) = 2 x + 2 x y + y and 2 x + y = 100 Gradient vector of a function f is expressed as: ∇ f ( x , y ) = f x ( x , y ) i + f y ( x , y ) j The partial derivative of a function f ( x , y ) = 2 x + 2 x y + y with respect to x is given as: f x ( x , y ) = ∂ ∂ x ( 2 x + 2 x y + y ) = 2 ∂ x ∂ x + 2 y ∂ x ∂ x + 0 = 2 + 2 y Partial derivative of function f ( x , y ) = 2 x + 2 x y + y with respect to y is given as: f y ( x , y ) = ∂ ∂ y ( 2 x + 2 x y + y ) = 2 ( 0 ) + 2 x ∂ y ∂ y + ∂ y ∂ y = 2 x + 1 Therefore, value of gradient vector ∇ f ( x , y ) is given as: ∇ f ( x , y ) = 2 x i − 2 y j Gradient vector of function g is expressed as: ∇ g ( x , y ) = g x ( x , y ) i + g y ( x ,

11th Edition

ISBN: 9781337275347

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 13.10, Problem 7E

To determine

To calculate: The maximum value of f(x,y)=2x+2xy+y using Lagrange multiplier, subject to constraint 2x+y=100 assuming x and y be positive.

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Chapter 13.10, Problem 6E

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The maximum value of f ( x , y ) = 2 x + 2 x y + y using Lagrange multiplier, subject to constraint 2 x + y = 100 assuming x and y be positive.

Solution Summary: The author explains how to calculate the maximum value of f(x,y) using Lagrange multiplier, subject to constraint 2x+y=100.

To calculate: The maximum value of f(x,y)=2x+2xy+y using Lagrange multiplier, subject to constraint 2x+y=100 assuming x and y be positive.

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Chapter 13 Solutions