A firm produces two products, A and B. The inverse demand functions are respectively, PA = 600 - 2x - 2y and PB = 1200 - 2x - - 4y, which are indeed prices of A and B per unit, where the firm produces and sells x units of product A and y units of product B. Its costs are given by CA = 5000+ 100x + x and CB = 9000+ 6y. (a) Show that the firm's total profit is given by: (x, y) = -3x - 10y - 4xy + 500x + 1200y - 14000. (b) Assume that (x, y) given by (a) has a maximum point. Find, step by step, the production levels (i.e. values of x and y) that maximise the profit by solving the first-order conditions (If you need to solve any system of linear equations, use Cramer's rule and provide all calcula- tion details). (c) Due to technology constraints, the total production must be restricted to be exactly 55 units. Find, step by step, the production levels that now maximise the profit - 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. (d) Report the Lagrange multiplier value at the maximum point; and the maximal profit value from part (c). No explanation is needed. (e) Using new technology, the total production can now be up to 120 units (i.e. less than or equal to 120 units). Use the values from (c) and (d) to approximate the new maximal profit. (f) Calculate the true new maximal profit for part (e) and compare with its approximate value you obtained. By what percentage is the true maximal profit different from the approximate value?