A monopolist has a demand function p=152,5−3q, and costs given by TVC=0,5q3−15q2+175q and TFC=0. Find the loss profits points. a. 0<q<2 and q>22 b. 2<q<22 and q>0 c. q>0 and q<2 d. q<22
Question
A monopolist has a demand function p=152,5−3q, and costs given by TVC=0,5q3−15q2+175q and TFC=0. Find the loss profits points.
a. 0<q<2 and q>22
b. 2<q<22 and q>0
c. q>0 and q<2
d. q<22
Solution
The profit of a firm is given by the difference between total revenue (TR) and total cost (TC). The total revenue is given by the price (p) times the quantity (q), and the total cost is the sum of total variable cost (TVC) and total fixed cost (TFC).
First, let's find the total revenue function by substituting p from the demand function into the total revenue function:
TR = p*q = (152.5 - 3q) * q = 152.5q - 3q^2
Next, let's find the total cost function. Since TFC = 0, the total cost is just the total variable cost:
TC = TVC = 0.5q^3 - 15q^2 + 175q
Now, let's find the profit function by subtracting total cost from total revenue:
Profit = TR - TC = (152.5q - 3q^2) - (0.5q^3 - 15q^2 + 175q) Profit = -0.5q^3 + 18q^2 - 22.5q
Now, let's find the loss profit points by setting the profit function equal to zero and solving for q:
-0.5q^3 + 18q^2 - 22.5q = 0
This is a cubic equation, and solving it requires either factoring (if possible) or using numerical methods. However, the question provides intervals for q, so we can check which intervals result in a negative profit (which would indicate a loss).
a. For 0<q<2 and q>22, substitute q = 1 and q = 23 into the profit function. If either results in a negative profit, then that interval contains a loss profit point.
b. For 2<q<22 and q>0, substitute q = 3 and q = 1 into the profit function. If either results in a negative profit, then that interval contains a loss profit point.
c. For q>0 and q<2, substitute q = 1 into the profit function. If it results in a negative profit, then that interval contains a loss profit point.
d. For q<22, substitute q = 21 into the profit function. If it results in a negative profit, then that interval contains a loss profit point.
Without specific numerical solutions to the cubic equation, we can't definitively identify the loss profit points, but this process will identify the intervals in which they occur.
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