Explanation Given: The company can manufacture P ( x , y ) units of a certain product. The labor and the capital is used x units and y units respectively. The unit cost of the labor and the capital is p and q respectively. The allocated units by the management to manufacture the product is k units. Here x ∗ and y ∗ are the units of labor and capital and the terms P x ( x ∗ , y ∗ ) and P y ( x ∗ , y ∗ ) are non-zeros. 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 . Proof: From given, p and q are the unit cost of labor and capital in dollars and p ∗ and q ∗ the unit cost of labor and capital with allocated C dollars. Since p is the unit cost of labor, then px be the constrained value of labor. Similarly q is the unit cost of capital, then qy be the constrained value of capital The constraint equation P ( x ∗ , y ∗ ) in the form g ( x , y ) = p x + q y − k = 0 . The Lagrangian function is as follows. F ( x , y , λ ) = P ( x , y ) + λ g ( x , y ) = P ( x ∗ , y ∗ ) + λ ( p x + q y − k ) The above expression is called the Lagrange function while λ be the Lagrange multipliers. Solve the system of equations of function F to find critical points as follows. Differentiate F ( x , y , λ ) = P ( x ∗ , y ∗ ) + λ ( p x + q y − k ) with respect to x , d d x ( F ( x , y , λ ) ) = d d x ( P ( x ∗ , y ∗ ) + λ ( p x + q y − k ) ) F x = d d x ( P ( x ∗ , y ∗ ) ) + d d x ( λ ( p x + q y − k ) ) = P x ( x ∗ , y ∗ ) + λ p Differentiate F ( x , y , λ ) = P ( x ∗ , y ∗ ) + λ ( p x + q y − k ) with respect to y , d d y ( F ( x , y , λ ) ) = d d y ( P ( x ∗ , y ∗ ) + λ ( p x + q y − k ) ) F y

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Chapter 8.5, Problem 36E

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To show: The minimized level production Px(x∗,y∗)Py(x∗,y∗)=pq by method of Lagrange multipliers.

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Chapter 8.5, Problem 34E

Chapter 8.5, Problem 38E

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A closed rectangular box with a volume V cm3 . The cost of materials used in the box is a sen/cm2 for the top and the bottom, b sen/cm2 for front and back, and c sen/cm2 for the remaining sides. Find the dimensions of the box so that the cost of materials is minimized.

1.A box with a square base and open top should have a volume of 50cm3. Using Lagrange multipliers find the dimensions of the box that minimize the amount of material to be used.

1 Use the Lagrange multipliers to find the maximum and minimum values of f(x,y)=2x+y−2zf(x,y)=2x+y−2z subject to the constraint x2+y2+z2=900.x2+y2+z2=900. Also give the points where these extreme values occur. Maximum = Enter your answer; maximumEnter your answer; Minimum = Enter your answer; minimumEnter your answer; Maximum is at (Enter your answer; x-coordinate of the maximum point Enter your answer; , Enter your answer; y-coordinate of the maximum pointEnter your answer; , Enter your answer; z-coordinate of the maximum pointEnter your answer; ) Minimum is at (Enter your answer; x-coordinate of the minimum point Enter your answer; , Enter your answer; y-coordinate of the minimum pointEnter your answer; , Enter your answer; z-coordinate of the minimum pointEnter your answer; )

Chapter 8 Solutions

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Ch. 8.1 - What is a function of two variables? Give an...Ch. 8.1 - Prob. 2CQ Ch. 8.1 - Prob. 3CQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Let f(x, y) = x2+ 2xy x + 3. Compute f(1, 2),...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 - Let g(r, s, t) = res/t. Compute g(1, 1, 1), g(1,...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 - In Exercises 11-18, find the domain of the...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 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - In Exercises 19-24, sketch the level curves of the...Ch. 8.1 - Prob. 22E Ch. 8.1 - In Exercises 1924, 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 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - Prob. 33E Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - POISEITLLES LAW Poiseuilles Law states that the...Ch. 8.1 - COST FUNCTION FOR A LOUDSPEAKER SYSTEM Acrosonic...Ch. 8.1 - Prob. 38E Ch. 8.1 - REVENUE FUNCTIONS FOR DESKS Country Workshop...Ch. 8.1 - Prob. 40E Ch. 8.1 - 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The minimized level production P x ( x ∗ , y ∗ ) P y ( x ∗ , y ∗ ) = p q by method of Lagrange multipliers.

Solution Summary: The author explains how Lagrange multipliers can be used to obtain the relative extremum of the function f(x,y,lambda).

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To show: The minimized level production Px(x∗,y∗)Py(x∗,y∗)=pq by method of Lagrange multipliers.

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Chapter 8 Solutions