(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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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