Final answer:
To find the production level that maximizes profit, set the derivative of total revenue equal to the derivative of total cost, which equates marginal revenue with marginal cost. Solve for the quantity x that satisfies this condition, which is the profit-maximizing output level. Calculate profit by subtracting total cost from total revenue at this output level.
Step-by-step explanation:
The student has asked to calculate the production level that maximizes profit given a cost function c(x) and a demand function p(x). To find the profit-maximizing output, we must determine where the marginal revenue (MR) equals the marginal cost (MC). This is done by setting the derivative of the total revenue function (price times quantity) equal to the derivative of the cost function, essentially looking for where MR = MC.
First, we need to determine the total revenue function (TR = price x quantity). The price can be determined from the demand function p(x) = 3000 - 6x. Total revenue is then calculated by multiplying this price function by the quantity x, leading to the total revenue function TR(x) = x(3000 - 6x). Then we find the derivative of the TR function to get the marginal revenue.
Afterwards, we need to find the derivative of the cost function c(x) to determine the marginal cost. Once the MR and MC functions are identified, the quantity x at which they are equal gives us the profit-maximizing production level. This involves solving the equation formed by equating MR and MC for the variable x.
Calculating profit can be done by subtracting the total cost from the total revenue at the profit-maximizing level of output. Graphically, this can be represented as the maximum vertical distance between the total revenue curve and the total cost curve.