A firm has the following total-cost and demand functions: C = aQ^3 - bQ^2 + cQ + d Q = e - P (d) Find optimizing level of Q. (e) Chooses a,b,c,d and e such that there is only one profit-maximizing level of output Q. I found 2 solutions for question d, but in a very long and messy form (full of the variables a-e. I do not know how to simplify my answer). Thus, I am not able to do question e). However, by analysing the marginal cost function, I found that a>0, b>0, c>0, d>0 and b^2 < 3ac. C = aQ^3 - bQ^2 + cQ + d MC = dC/dQ = 3aQ^2 - 2bQ + c The coefficient of Q^2 must be positive, in order for the cost function to be U-shaped (MC must be U-shaped to make economic sense). Thus, a>0. MC' = dMC/dQ^2 = 6aQ - 2b = 0 Hence, Q = b/3a As Q must be positive, and a is positive, b must necessarily be positive: b>0 MCmin = 3a(b/3a)^2 - 2b(b/3a) + c = b^2/3a - 2b^2/3a + c = -b^2/3a + c =(-b^2 + 3ac)/3a thus, b^2 < 3ac and c > 0 d > 0 in order to make economic sense (it is a fixed cost). I also found the profit function = eQ - Q^2 - aQ^3 + bQ^2 - cQ - d and its derivative = e - 2Q - 3aQ^2 + 2bQ - c = 0 and solved for q q = (-2b-2 +/- root(4b^2 - 8b + 4 + 12ae - 12ac)) / -6a Unfortunately, from here on, I'm stuck. Any advice?