Calculus question

Question

the marginal cost of a product can be thought of as the cost of producing one additional unit of output. for ex: if the marg. cost of producing the 50th product is $6.20, it cost $6.20 to increase production from 49 to 50 units of output. Suppose the marginal cost C (in dollars) produce x thousand mp3 players is given by the function C(x)=x^2-120x+750

a. how many players should be produced to minimize the marginal cost?

b. what is the minimum marginal cost?

Answer 1

The marginal cost is the derivative of the cost function. So, when we are being asked to minimize the marginal cost (minimize the derivative of the cost function), we are being asked to set the second derivative equal to zero and solve.

In this case however, they have given us the marginal cost function (which already is the first derivative of the cost function) instead of the cost function so we just need to take the derivative of that marginal cost function and set it equal to zero.

To avoid confusion, I'm going to define the marginal cost function as MC(x) [instead of C(x)]

MC(x) = x² - 120x + 750

MC '(x) = 2x - 120

Setting it equal to zero, we have 2x - 120 = 0 or 2x = 120 or x = 60

So, producing 60 units gives the minimum marginal cost.

Since the marginal cost is defined by MC(x) = x² - 120x + 750, then the minimum MC is:

MC(60) = (60)² - 120(60) + 750 = 3600 - 7200 + 750 = -$2850/unit