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(a) A firm produces two product brands, the quantities of which are x1 and x2. The demand for brand x2 is three times that for brand x1. The profit function of the firm is T (x1, x2) = 30x1 – 3x² + 5x1x2 – 2x² + 10x2. %3D The firm wishes to maximize its profit. (i) (ii) (iii) Formulate the problem as a constrained optimization problem Write down the Lagrange function. Write down the first-order conditions.

(a) A firm produces two product brands, the quantities of which are x1 and x2. The demand for brand x2 is three times that for brand x1. The profit function of the firm is T (x1, x2) = 30x1 – 3x² + 5x1x2 – 2x² + 10x2. %3D The firm wishes to maximize its profit. (i) (ii) (iii) Formulate the problem as a constrained optimization problem Write down the Lagrange function. Write down the first-order conditions.

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.8: Linear Programming
Problem 3SC: In Example 3, if the accountant earns a profit of 100 on each individual return and a profit of 175...
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(a) A firm produces two product brands, the quantities of which are x1 and x2. The demand for brand x2 is three times that for brand x1. The profit function of the firm is T (x1, x2) = 30x1 – 3x² + 5x1x2 – 2x² + 10x2. %3D The firm wishes to maximize its profit. (i) (ii) (iii) (iv) Formulate the problem as a constrained optimization problem Write down the Lagrange function. Write down the first-order conditions. Solve for the optimal values of x1, X2 and the Lagrange multiplier. (v) (vi) What is the maximum profit? Use the bordered Hessian to prove that the profit is indeed maximized.
Transcribed Image Text:(a) A firm produces two product brands, the quantities of which are x1 and x2. The demand for brand x2 is three times that for brand x1. The profit function of the firm is T (x1, x2) = 30x1 – 3x² + 5x1x2 – 2x² + 10x2. %3D The firm wishes to maximize its profit. (i) (ii) (iii) (iv) Formulate the problem as a constrained optimization problem Write down the Lagrange function. Write down the first-order conditions. Solve for the optimal values of x1, X2 and the Lagrange multiplier. (v) (vi) What is the maximum profit? Use the bordered Hessian to prove that the profit is indeed maximized.
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