Explanation Given: The utility function f ( x , y ) = x y 2 and the cost of unit of x is $1, the cost of a unit of y $2 and available amount is $60. Definition used: Lagrange Multipliers 1. The form of the constraint is g ( x , y ) = 0 . 2. The form of the Lagrange function F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) . 3 Form the system of equations F x ( x , y , λ ) = 0 , F y ( x , y , λ ) = 0 and F λ ( x , y , λ ) = 0 , compute the solution of these system and also the relative extrema among these solution. Calculation: The function f ( x , y ) = x y 2 . Here, the available amount is $60, the cost of x is $1 and the cost of y is $2. x + 2 y = 60 The function f ( x , y ) = x y 2 is subject to the constraint x + 2 y = 60 . The constrain in the form g ( x , y ) = 0 . Thus, the given constrain can be written as g ( x , y ) = x + 2 y − 60 . The form of the Lagrange function is, F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) . Substitute f ( x , y ) = x y 2 and g ( x , y ) = x + 2 y − 60 in F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) . F ( x , y , λ ) = x y 2 − λ ( x + 2 y − 60 ) = x y 2 − λ x − 2 λ y + 60 λ Thus, the Lagrange function is F ( x , y , λ ) = x y 2 − λ x − 2 λ y + 60 λ . Need to find the partial derivative of the function F ( x , y , λ ) . The Partial derivative of the function F ( x , y , λ ) with respect to x and fix y and λ are constant. F x ( x , y , λ ) = ∂ ∂ x ( x y 2 − λ x − 2 λ y + 60 λ ) = ∂ ∂ x ( x y 2 ) − ∂ ∂ x ( λ x ) − ∂ ∂ x ( 2 λ y ) + ∂ ∂ x ( 60 λ ) = ( y 2 ) ∂ ∂ x ( x ) − ( λ ) ∂ ∂ x ( x ) − ∂ ∂ x ( 2 λ y ) + ∂ ∂ x ( 60 λ ) Simplify the terms and obtain the partial derivatives, F x ( x , y , λ ) = ( y 2 ) ( 1 ) − ( λ ) ( 1 ) − 0 + 0 = y 2 − λ Therefore, the partial derivative with respect to x is F x ( x , y , λ ) = y 2 − λ . The Partial derivative of the function F ( x , y , λ ) with respect to y and fix x , z and λ are constant. F y ( x , y , λ ) = ∂ ∂ y ( x y 2 − λ x − 2 λ y + 60 λ ) = ∂ ∂ y ( x y 2 ) − ∂ ∂ y ( λ x ) − ∂ ∂ y ( 2 λ y ) + ∂ ∂ y ( 60 λ ) = ( x ) ∂ ∂ y ( y 2 ) − ∂ ∂ y ( λ x ) − ( 2 λ ) ∂ ∂ y ( y ) + ∂ ∂ y ( 60 λ ) Simplify the terms and obtain the partial derivatives, F y ( x , y , λ ) = ( x ) ( 2 y ) − 0 − ( 2 λ ) ( 1 ) + 0 = 2 x y − 2 λ Therefore, the partial derivative with respect to x is F y ( x , y , z , λ ) = 2 x y − 2 λ . The Partial derivative of the function F ( x , y , λ ) with respect to λ and fix x and y are constant. F λ ( x , y , λ ) = ∂ ∂ λ ( x y 2 − λ x − 2 λ y + 60 λ ) = ∂ ∂ λ ( x y 2 ) − ∂ ∂ λ ( λ x ) − ∂ ∂ λ ( 2 λ y ) + ∂ ∂ λ ( 60 λ ) = ∂ ∂ λ ( x y 2 ) − ( x ) ∂ ∂ λ ( λ ) − ( 2 y ) ∂ ∂ λ ( λ ) + ( 60 ) ∂ ∂ λ ( λ ) Simplify the terms and obtain the partial derivatives, F λ ( x , y , λ ) = 0 − ( x ) ( 1 ) − ( 2 y ) ( 1 ) + ( 60 ) ( 1 ) = − x − 2 y + 60 Therefore, the partial derivative with respect to z is F λ ( x , y , λ ) = − x − 2 y + 60

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

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Chapter 17.4, Problem 23E

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To maximize: The utility functions with the cost of each commodity.

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Chapter 17.4, Problem 22E

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Chapter 17 Solutions

Finite Mathematics and Calculus with Applications (10th Edition)

Ch. 17.1 - For the function in Example 1, find f(2, 3)....Ch. 17.1 - Prob. 2YT Ch. 17.1 - Prob. 3YT Ch. 17.1 - Prob. 4YT Ch. 17.1 - Prob. 1WE Ch. 17.1 - Prob. 2WE Ch. 17.1 - Prob. 3WE Ch. 17.1 - Prob. 4WE Ch. 17.1 - Prob. 5WE Ch. 17.1 - Prob. 6WE

Ch. 17.1 - Let f(x, y) = 2x 3y + 5. Find the following....Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E Ch. 17.1 - Prob. 7E Ch. 17.1 - Prob. 8E Ch. 17.1 - Graph the first-octant portion of each plane. 9.2x...Ch. 17.1 - Prob. 10E Ch. 17.1 - Prob. 11E Ch. 17.1 - Prob. 12E Ch. 17.1 - Prob. 13E Ch. 17.1 - Prob. 14E Ch. 17.1 - Prob. 15E Ch. 17.1 - Prob. 16E Ch. 17.1 - Prob. 17E Ch. 17.1 - Prob. 18E Ch. 17.1 - Prob. 19E Ch. 17.1 - Prob. 20E Ch. 17.1 - Prob. 21E Ch. 17.1 - Prob. 22E Ch. 17.1 - Match each equation in Exercises 2328 with its...Ch. 17.1 - Prob. 24E Ch. 17.1 - Prob. 25E Ch. 17.1 - Match each equation in Exercises 2328 with its...Ch. 17.1 - Prob. 27E Ch. 17.1 - Prob. 28E Ch. 17.1 - Prob. 29E Ch. 17.1 - Prob. 30E Ch. 17.1 - Prob. 31E Ch. 17.1 - Prob. 32E Ch. 17.1 - Prob. 33E Ch. 17.1 - Prob. 34E Ch. 17.1 - Production Find the level curve at a production of...Ch. 17.1 - Prob. 36E Ch. 17.1 - Prob. 37E Ch. 17.1 - Individual Retirement Accounts The multiplier...Ch. 17.1 - Individual Retirement Accounts The multiplier...Ch. 17.1 - Heat Loss The rate of heal loss (in watts) in...Ch. 17.1 - Prob. 41E Ch. 17.1 - Prob. 42E Ch. 17.1 - Pollution Intolerance According to research at the...Ch. 17.1 - Dengue Fever In tropical regions, dengue fever is...Ch. 17.1 - Prob. 45E Ch. 17.1 - Prob. 46E Ch. 17.1 - Prob. 47E Ch. 17.1 - Prob. 48E Ch. 17.1 - Prob. 49E Ch. 17.1 - Prob. 50E Ch. 17.1 - Prob. 51E Ch. 17.2 - Prob. 1YT Ch. 17.2 - Prob. 2YT Ch. 17.2 - Prob. 3YT Ch. 17.2 - Prob. 4YT Ch. 17.2 - Prob. 1WE Ch. 17.2 - Prob. 2WE Ch. 17.2 - Prob. 3WE Ch. 17.2 - Prob. 4WE Ch. 17.2 - Prob. 5WE Ch. 17.2 - Prob. 6WE Ch. 17.2 - Prob. 7WE Ch. 17.2 - Prob. 8WE Ch. 17.2 - Prob. 1E Ch. 17.2 - Prob. 2E Ch. 17.2 - Prob. 3E Ch. 17.2 - Prob. 4E Ch. 17.2 - Prob. 5E Ch. 17.2 - Prob. 6E Ch. 17.2 - Prob. 7E Ch. 17.2 - Prob. 8E Ch. 17.2 - Prob. 9E Ch. 17.2 - Prob. 10E Ch. 17.2 - Prob. 11E Ch. 17.2 - Prob. 12E Ch. 17.2 - Prob. 13E Ch. 17.2 - Prob. 14E Ch. 17.2 - Prob. 15E Ch. 17.2 - Prob. 16E Ch. 17.2 - 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Prob. 61E Ch. 17.2 - Prob. 62E Ch. 17.2 - Drug Reaction The reaction to x units of a drug t...Ch. 17.2 - Prob. 64E Ch. 17.2 - Prob. 66E Ch. 17.2 - Prob. 67E Ch. 17.2 - Prob. 68E Ch. 17.2 - Gravitational Attraction The gravitational...Ch. 17.2 - Velocity In 1931, Albert Einstein developed the...Ch. 17.2 - Prob. 71E Ch. 17.3 - Prob. 1YT Ch. 17.3 - Prob. 2YT Ch. 17.3 - Prob. 1WE Ch. 17.3 - Prob. 2WE Ch. 17.3 - Prob. 1E Ch. 17.3 - Prob. 2E Ch. 17.3 - Prob. 3E Ch. 17.3 - Prob. 4E Ch. 17.3 - Prob. 5E Ch. 17.3 - Prob. 6E Ch. 17.3 - Prob. 7E Ch. 17.3 - Prob. 8E Ch. 17.3 - Prob. 9E Ch. 17.3 - Prob. 10E Ch. 17.3 - Prob. 11E Ch. 17.3 - Prob. 12E Ch. 17.3 - Prob. 13E Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - Prob. 16E Ch. 17.3 - Prob. 17E Ch. 17.3 - Prob. 18E Ch. 17.3 - Prob. 19E Ch. 17.3 - Prob. 20E Ch. 17.3 - Prob. 21E Ch. 17.3 - Prob. 22E Ch. 17.3 - Prob. 23E Ch. 17.3 - Figures (a)(f) show the graphs of the functions...Ch. 17.3 - Prob. 25E Ch. 17.3 - Prob. 26E Ch. 17.3 - Prob. 27E Ch. 17.3 - Prob. 28E Ch. 17.3 - Prob. 29E Ch. 17.3 - Prob. 30E Ch. 17.3 - Prob. 31E Ch. 17.3 - Prob. 32E Ch. 17.3 - Prob. 33E Ch. 17.3 - Prob. 34E Ch. 17.3 - Prob. 35E Ch. 17.3 - Prob. 36E Ch. 17.3 - Prob. 37E Ch. 17.3 - Prob. 38E Ch. 17.3 - Prob. 39E Ch. 17.3 - Prob. 40E Ch. 17.3 - Prob. 42E Ch. 17.4 - Prob. 1YT Ch. 17.4 - Prob. 1WE Ch. 17.4 - Prob. 2WE Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Prob. 11E Ch. 17.4 - Find positive numbers x and y such that x + y = 48...Ch. 17.4 - 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If the function...Ch. 17 - Prob. 85RE Ch. 17 - Prob. 86RE Ch. 17 - Prob. 87RE Ch. 17 - Prob. 88RE Ch. 17 - Prob. 89RE Ch. 17 - Prob. 90RE Ch. 17 - Prob. 91RE Ch. 17 - Prob. 92RE Ch. 17 - Prob. 93RE Ch. 17 - Prob. 94RE Ch. 17 - Prob. 95RE Ch. 17 - Prob. 96RE Ch. 17 - Profit The total profit from 1 acre of a certain...Ch. 17 - Prob. 98RE Ch. 17 - Prob. 99RE Ch. 17 - Prob. 100RE Ch. 17 - Prob. 101RE Ch. 17 - Prob. 102RE Ch. 17 - Prob. 103RE Ch. 17 - Prob. 104RE Ch. 17 - Prob. 105RE Ch. 17 - Prob. 106RE

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To maximize: The utility functions with the cost of each commodity.

Solution Summary: The author explains the maximization of utility functions with the cost of each commodity. The form of the constraint is g(x,y)=0.

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Chapter 17 Solutions