# Profit function and output for maximum profit

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?

• Never put useless, irrelevant phrases such as "Is this correct" in a title. – David G. Stork Mar 23 '18 at 1:19
• It seems right to. Only $Q^*=\sqrt{66}-1$ is a valid solution. – callculus Mar 23 '18 at 5:26