If the demand function of the monopolist is {p=200-0.5x} and its cost function is {C=100+5x+7x^2}. Value of 'x' , price and the amount of profit for which profit maximizes are:
Not really familiar with the formulas relating demand function, cost functions and profit, but here is an example of your question.
http://math.stackexchange.com/questions/758620/maximized-profit-w-a-cost-demand-function
thanks
To find the value of 'x' at which the monopolist maximizes profit, we need to determine the quantity where marginal cost equals marginal revenue.
First, let's find the marginal cost (MC) and marginal revenue (MR) functions:
Given the cost function C = 100 + 5x + 7x^2, we can calculate the marginal cost as the derivative of the cost function with respect to quantity (x):
MC = dC/dx = 5 + 14x
Next, the marginal revenue can be calculated by taking the derivative of the demand function with respect to quantity:
MR = d(p*x)/dx = d(200x - 0.5x^2)/dx = 200 - x
Now, we need to find the quantity (x) at which MC equals MR:
MC = MR
5 + 14x = 200 - x
Simplifying the equation:
15x = 195
x = 13
Therefore, the value of 'x' at which the monopolist maximizes profit is 13.
To calculate the price (p) and profit at this quantity, we substitute 'x' into the original demand function:
p = 200 - 0.5x
p = 200 - 0.5 * 13
p = 193.5
So, the price at this quantity is 193.5.
To calculate the profit, we subtract the total cost (C) from the total revenue (R). The total revenue (TR) is given by p*x, and the total cost (TC) is given by the cost function C:
TR = p*x = 193.5 * 13
TR = 2515.5
TC = C = 100 + 5x + 7x^2
TC = 100 + 5*13 + 7*13^2
TC = 100 + 65 + 1183
TC = 1348
Profit (Ξ ) = TR - TC
Profit = 2515.5 - 1348
Profit β 1167.5
Therefore, the amount of profit that maximizes profit is approximately 1167.5.
profit = revenue - cost
revenue = price * demand
P(x) = x(200-0.5x) - (100+5x+7x^2)
= -7.5x^2 + 195x - 100
Now just find the vertex of that parabola.