Maximize profit

Name: Brian

Who is asking: Student Level: Secondary

Question:
The marginal cost for a certain product is given by MC = 6x+60 and the fixed costs are $100. The marginal revenue is given by MR = 180-2x. Find the level of production that will maximize profit and find the profit or loss at that level.

Hi Brian,

If C(x) is the cost function where x is the level of production of the product being produced them the marginal cost, MC(x), is the derivative of C(x). Thus, using antidifferentiation you can find the cost function. Since

MC(x) = C'(x) = 6 x + 60 antidifferentiation gives C(x) = 3 x² + 60 x + K where K is some constant.

You are also told that the the fixed costs are $100, in other words C(0) = $100. Thus

C(0) = 3(0)² + 60(0) + K = 100 Thus K = 100 and hence C(x) = 3 x² + 60 x + 100

In a similar way you can find the revenue function, M(x), since M'(x) = MR(x). In this case, if you produce no product the revenue is $0 and hence R(0) = 0.

The profit function, P(x), is the revenue the cost, and hence

P(x) = R(x) - C(x). Thus to maximize the profit, find P'(x) and solve P'(x) = 0. Notice that the equation you have to solve is MC(x) = MR(x)

I hope this helps,
Harley Go to Math Central