The optimization problem is the hint to the main solution. I would appreciate having a solution written with nice handwriting and detail explanations.

A firm can produce its product in two plants. The cost of producing output qi in Plant i is ,cig for some positive constant ci. In addition, Plant i can produce at most qi output. The market price of the product is (1 - q1 - q2) where (q1 + 92) is the total output. Find the firm's optimal production plan. Ignore the transportation cost. Solution. We solve the following program: haxqi,92 (q1 + q2) (1 - q1 - 92) - 20197 - 20293; s.t. qi _ qi, for i = 1, 2. Let Mi be the Lagrange multiplier of the constraint. The optimality conditions are 1 - 2(q1 + q2) - ciqi - Xi = 0, for i = 1, 2; di 2 0; qi