Explanation Given: The total of ten kits must be made. 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: Let x be a large jewelry-making kits and y be a small jewelry-making kits. Since the total of ten kits must be made. x + y = 10 The function C ( x , y ) = 2 x 2 + 6 y 2 + 4 x y + 10 subject to the constraint x + y = 10 . The constraint is in the form g ( x , y ) = 0 . Thus, the given constrain can be written as g ( x , y ) = x + y − 10 . The form of the Lagrange function is, F ( x , y , λ ) = C ( x , y ) − λ ⋅ g ( x , y ) . Substitute C ( x , y ) = 2 x 2 + 6 y 2 + 4 x y + 10 and g ( x , y ) = x + y − 10 in F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) , F ( x , y , λ ) = 2 x 2 + 6 y 2 + 4 x y + 10 − λ ( x + y − 10 ) = 2 x 2 + 6 y 2 + 4 x y + 10 − λ x − λ y + 10 λ Thus, the Lagrange function is F ( x , y , λ ) = 2 x 2 + 6 y 2 + 4 x y + 10 − λ x − λ y + 10 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 ( 2 x 2 + 6 y 2 + 4 x y + 10 − λ x − λ y + 10 λ ) = ∂ ∂ x ( 2 x 2 ) + ∂ ∂ x ( 6 y 2 ) + ∂ ∂ x ( 4 x y ) + ∂ ∂ x ( 10 ) − ∂ ∂ x ( λ x ) − ∂ ∂ x ( λ y ) + ∂ ∂ x ( 10 λ ) = ( 2 ) ∂ ∂ x ( x 2 ) + ∂ ∂ x ( 6 y 2 ) + ( 4 y ) ∂ ∂ x ( x ) + ∂ ∂ x ( 10 ) − ( λ ) ∂ ∂ x ( x ) − ∂ ∂ x ( λ y ) + ∂ ∂ x ( 10 λ ) Simplify the terms and obtain the partial derivatives, F x ( x , y , λ ) = ( 2 ) ( 2 x ) + 0 + ( 4 y ) ( 1 ) + 0 − ( λ ) ( 1 ) − 0 + 0 = 4 x + 4 y − λ Therefore, the partial derivative with respect to x is F x ( x , y , λ ) = 4 x + 4 y − λ

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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Chapter 17.4, Problem 29E

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Chapter 17.4, Problem 28E

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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 - 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The reason for how should production be allocated if the total cost is minimized.

Solution Summary: The author explains how production should be allocated if the total cost is minimized. The large jewelry-making kit is 10 and no small kits to maximum profit.

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Chapter 17 Solutions