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) = 0.1X 0.2xy 0.2y + 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.