Explanation:
A. To find the profit-maximizing level of price and output, we need to find the equilibrium point where the marginal revenue equals marginal cost.
The marginal cost of production is the derivative of the total cost function: MC = dC/dQ = 3.
The marginal revenue is the derivative of the inverse demand function: MR = dP/dQ = -1.
So at the profit-maximizing level of output, MC = MR.
3 = -1
Q = 6
Then we can substitute Q = 6 back into the inverse demand function to find the price:
P (Q) = 12 - Q
P (6) = 12 - 6
P = 6
So the profit-maximizing level of output is 6 units at a price of $6.
B. To find the maximum profit of the firm, we need to calculate the total revenue and the total cost at the profit-maximizing level of output.
The total revenue is the product of the price and the quantity:
TR = P * Q
TR = 6 * 6
TR = 36
The total cost is given by the total cost function:
TC = C (Q)
TC = 3 * Q
TC = 3 * 6
TC = 18
So the maximum profit is the difference between the total revenue and the total cost:
Profit = TR - TC
Profit = 36 - 18
Profit = 18
The maximum profit of the firm is $18 at an equilibrium price level of $6 and an output level of 6 units.