Optimization with calculus

Question

If R(x) is the revenue that a company receives when it sells x units of a product, then the marginal revenue function is the derivative R'(x). The profit function is P(x) = R(x) − C(x).

(a) Show that if the profit P(x) is a maximum, then the marginal revenue equals the marginal cost.

We are given that the total profit is P(x) = R(x) − C(x). In order to maximize profit we look for the critical exponential points of P, that is, the numbers where the marginal profit is 0. But if P'(x) = R'(x) − C'(x) then R'(x) = C'(x). Therefore, if the profit is a maximum, then the marginal revenue equals the marginal cost.

(b) If C(x) = 16,000 + 400x − 1.6x² + 0.004x³

 is the cost function and p(x) = 1600 − 7x

 is the demand function, find the production level that will maximize profit.

Answer 1

Hi Mira K.,

So what is the part of the problem that you need help on?

You know that you have to take the first derivative of each of those equations (for C(x) and P(x)) with respect to x. So do that. Then set the resultant expressions (they're polynomials) equal to each other, and solve for x. That's your answer (units of production which will optimize profit). Hopefully (if you are the manufacturer) the result is a positive number!!!

-- Cheers, -- Mr. d.