The profit obtained by a firm from producing and selling x and y units of two brands of a commodity is given by
P(x, y) = ?0.1X² ? 0.2xy ? 0.2y² + 47x + 48y ? 600.
(a) Assume P(x, y) has a maximum point. Find, step by step, the production levels that maximize profit by solving the first-order conditions. If you need to solve any system of linear equations, use Cramer’s rule and provide all calculation details.
(b) Due to technology constraints, the total production must be restricted to be 200 units. Find, step by step, the production levels that now maximize profits – using the Lagrange Method. If you need to solve any system of linear equations, use Cramer’s rule and provide all calculation details. You may assume that the optimal point exists in this case.
(c) Report the Lagrange multiplier value at the maximum point and the maximal profit value from question (b). No explanation is needed.
(d) Using new technology, the total production can now be up to 250 units. Use the values from question (c) to approximate the new maximal profit.
(e) Compare the true new maximal profit for question (d) with its approximate value you obtained. By what percentage is the true maximal profit larger than the approximate value from question (c)?