Suppose that a firm produces two outputs with cost function C where C(q) = q + 2q3 + 10 given output levels q = (q1, 92) > 0. You could disregard the nonnegativity constraints on q and think of this as an exercise of unconstrained optimization. (a) If the firm took the output prices as given and fixed, what would be the profit-maximizing solution(s)? Would the solution be unique? Explain. Now, for the rest of the problem (b)-(d), assume that the firm is a monopoly in both of the output markets. The demand functionss for the two outputs, Qi and Q2, satisfy Q1(P) = A1 - 2P aP2, Q2(P) = A2 - bP - 3P2 %3D for all output prices P = (P1, P2) > 0, where A1, A2, a, b are positive real numbers and are given exogenously. The firm takes these downward-sloping demand curves when choosing output levels. (b) Find the output levels that satisfy the first-order condition for maximum profit. (c) Check the second-order condition. What would you conclude from your finding? Is there a unique solution? How would you answer depend on A1, A2, a, b? (d) Compute the maximum profit level, assuming that the second-order condition holds for an interior solution.