The total cost of producing X units of a product by a perfectly competitive firm is given by the expression TC = 15 + 0.4*X + 0.1*X^2. The market price of each product is $10. If X units of the product are sold, the revenue is 10*X.
The profit earned by the firm when X units are sold is given by P = 10X - 15 - 0.4X - 0.1X^2. To maximize the profit the number of units that should by produced is equal to the solution of the equation P' = 0 where P' is the first derivative of P with respect to X.
P' = 10 - 0.4 - 0.2*X
10 - 0.4 - 0.2*X = 0
=> X = 48
The firm should produce 48 units to maximize the profit made. The maximum profit it can earn is $215.4