Q:  If a competitive firm has TC=q3/3+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
Therefore the optimal q is where q2=49, which is q=7.

Profits =TR-TC=PQ-TC=49*7-(73/3+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