The inverse of the demand function is; P = 9 - 0.25Q
The profit-maximizing price and quantity are; $8.5 and 2 units.
The maximum profit is; $1
How to find the inverse of a function?
A) The demand function we are given is;
Q = 36 - 4P
Making P the subject gives the inverse demand function;
P = (36 - Q)/4
P = 9 - Q/4
P = 9 - 0.25Q
B) The profit-maximization point is the point at which MR = MC.
MR refers to the marginal revenue and MC is the marginal cost.
MC can be calculated as the first derivative of the cost function:
C(Q) = 4 + 4Q + Q²
MC = C'(Q) = 2Q + 4
Total Revenue = Price * Quantity
Total Revenue = (9 - 0.25Q) * Q
Total Revenue = 9Q - 0.25Q²
MR is gotten by differentiating Total Revenue to get;
MR = 9 - 0.5Q
Applying the condition MR = MC, we have;
9 - 0.5Q = 4 - 2Q
Solving for Q gives Q = 2
Thus, profit maximizing quantity is 2.
Thus, profit maximizing price will be;
P(2) = 9 - 0.25(2)
P(2) = $8.5
C) Formula for Maximum Profit is;
Profit = Total Revenue - Total Cost
Total Revenue = 8.5 * 2
Total revenue = $17
Total Cost is;
C(2) = 4 + 4(2) + 2²
C(2) = $16
Thus;
Maximum Profit = 17 - 16 = $1
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