Explanation Result used: Let the partial derivatives f x x , f y y and f x y of the function z = f ( x , y ) exists in the circular region contained in the xy -plane with center ( a , b ) and f x ( a , b ) = 0 , f y ( a , b ) = 0 and the discriminant D = f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 . (a) The function has relative maximum at ( a , b ) if D > 0 and f x x ( a , b ) < 0 . (b) The function has relative minimum at ( a , b ) if D > 0 and f x x ( a , b ) > 0 . (c) If D < 0 , then f ( a , b ) is a saddle point (neither a maximum nor a minimum) (d) If D = 0 , then the test fail. Calculation: The given labor cost function is, C ( x , y ) = 2 x 2 + 2 y 2 − 3 x y + 4 x − 94 y + 4200 . Calculate the partial derivative of the above function with respect to x . C x ( x , y ) = ∂ ∂ x ( 2 x 2 + 2 y 2 − 3 x y + 4 x − 94 y + 4200 ) = ∂ ∂ x ( 2 x 2 ) + ∂ ∂ x ( 2 y 2 ) − ∂ ∂ x ( 3 x y ) + ∂ ∂ x ( 4 x ) − ∂ ∂ x ( 94 y ) + ∂ ∂ x ( 4200 ) = 2 ( 2 x ) + 0 − 3 y + 4 − 0 + 0 = 4 x + 4 − 3 y Again, take the partial derivative of the function C x ( x , y ) with respect to x . C x x ( x , y ) = ∂ ∂ x ( L x ( x , y ) ) = ∂ ∂ x ( 4 x + 4 − 3 y ) = ∂ ∂ x ( 4 x ) + ∂ ∂ x ( 4 ) − ∂ ∂ x ( 3 y ) = 4 That is, C x x ( x , y ) = 4 (1) Calculate the partial derivative of the function C ( x , y ) with respect to y . C y ( x , y ) = ∂ ∂ y ( 2 x 2 + 2 y 2 − 3 x y + 4 x − 94 y + 4200 ) = ∂ ∂ y ( 2 x 2 ) + ∂ ∂ y ( 2 y 2 ) − ∂ ∂ y ( 3 x y ) + ∂ ∂ y ( 4 x ) − ∂ ∂ y ( 94 y ) + ∂ ∂ y ( 4200 ) = 0 + 4 y − 3 x + 0 − 94 + 0 = 4 y − 3 x − 94 Again, take the partial derivative of the function C y ( x , y ) with respect to y . C y y ( x , y ) = ∂ ∂ y ( C y ( x , y ) ) = ∂ ∂ y ( 4 y − 3 x − 94 ) = ∂ ∂ y ( 4 y ) − ∂ ∂ y ( 3 x ) − ∂ ∂ y ( 94 ) = 4 That is, C y y ( x , y ) = 4 (2) Calculate the partial derivative of the function C x ( x , y ) with respect to y

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 17.3, Problem 36E

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To find: The values of x and y such that to minimize the total cost function C(x,y)=2x2+2y2−3xy+4x−94y+4200 and use the values x and y to find the minimum total cost.

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Chapter 17.3, Problem 35E

Chapter 17.3, Problem 37E

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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. 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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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The values of x and y such that to minimize the total cost function C ( x , y ) = 2 x 2 + 2 y 2 − 3 x y + 4 x − 94 y + 4200 and use the values x and y to find the minimum total cost.

Solution Summary: The author explains that the minimum value of the labor cost function is 1,832 and it is occurred when the production of electrical tape x=38 units and packing tape

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To find: The values of x and y such that to minimize the total cost function C(x,y)=2x2+2y2−3xy+4x−94y+4200 and use the values x and y to find the minimum total cost.

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Chapter 17 Solutions