Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. Over what interval in the domain do average costs increase?

Question

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by  ,  , where x is the number of machines used. Over what interval in the domain do average costs increase? ​

Quizplus · Accepted Answer

We can find the interval in which average costs increase by finding the derivative of the average cost function and determining when it is positive.

Taking the derivative of average costs with respect to the number of machines used, we get:

C'(x) = 16x - 136

Setting this equal to 0 and solving for x, we get:

16x - 136 = 0
x = 8.5

So, the derivative is 0 at x = 8.5. We can use the first derivative test to determine that the derivative is negative for x < 8.5 and positive for x > 8.5. Therefore, average costs increase for x > 8.5.

To find the interval in the domain of x where the average costs increase, we need to determine the domain of the function. Since we cannot have negative machines, the domain is from 0 to infinity.

The correct choice is B, since it represents the interval from 9 to infinity.

Suppose the Average Costs of a Mining Operation Depend on the Number