Q: If a competitive firm has TC=q^{3}/3+25 and faces a market price
of Pe=49, what is the firm's maximum profit?

A: The first step is to determine the optimal quantity which is where
MR=MC.

Since the firm is competitive MR=Pe=49.

MC=dTC/dQ=q^{2
}Therefore the optimal q is where q^{2}=49, which is q=7.

Profits =TR-TC=PQ-TC=49*7-(7^{3}/3+25) = 203.67.

The last step is to make sure that the firm prefers operating to shutting
down and paying the fixed cost.

In this case, the firm prefers to produce since profit>-FC=-25