Firm is a monopolist.
Demand Function: P = 210-5Q
Total Cost Function: TC = $ Q^3 - 2Q^2 + 15Q + 60$
Derive the profit function and calculate the output level to give maximum profit or minimum loss.
Profit = Total revenue - Total Costs
TR = PQ -> $210Q-5Q^2$
TC = $ Q^3 - 2Q^2 + 15Q + 60$
So profit = $195Q - 3Q^2 - 3Q^3 - 60$
Max profit level is when MR=MC
MR = $\frac{dTR}{dQ} = 210-10Q$
MC = $\frac{dTC}{dQ}$ = $ 3Q^2 - 4Q + 15$
Then make MR=MC and solve. I ended up with a quadratic formula, solving for $ Q = \sqrt66 -1$ or $ -1 - \sqrt66$
Can someone verify my answer and my thought process? Many thanks.
If my answer is correct, how would i calculate the price elasticity of demand as output with the maximum profit or minimum loss level? This surely would be impossible to do with imaginary numbers?