The document uses calculus to determine the profit-maximizing level of output for a firm in perfect competition facing a price of $4 and total costs of Q3 – 7Q2 + 12Q+5. It finds the marginal revenue is $4 and sets marginal cost equal to marginal revenue. This yields two possible output levels: 2/3 and 4. However, only at 4 is the slope of the marginal cost curve positive, indicating profit maximization. The total profit at this optimal output of 4 is calculated to be $11.

1. Example
A perfect competitive firm faces P=4 and TC = Q3 –
7Q2 + 12Q+5
Determine by using calculus the best level of out
put of the firm by the marginal approach.
TR = PQ = 4Q
MR = d(TR) = 4 = P
dq
MC = d(TC) = 3Q2 – 14Q +12
dq

2. 4 = 3Q2 – 14Q +12
0 = 3Q2 – 14Q +8
0 = (3Q – 2)(Q-4)
So that , Q = 2/3 and Q = 4
Thus, MR =MC at Q =2/3 and Q=4
In order for profit maximized rather than minimized, the MC
curve should be rising(slope must be positive) at the point where
MR =MC.
Slope of the MC curve = d(MC) = 6Q -14
dq
At Q=2/3, the slope is -10(the firm minimizes total profit)
At Q=4, the slope is 10 so the firm maximizes its total profits.

3. • Find the total profit of the firm at this level of
out put
Profit = TR –TC
= 4Q - 3Q2 – 14Q +12
= - Q3 – 7Q2 +8Q-5
= -64+112-32-5
= 11