Use the Second Derivative Test to determine the critical number that gives a local minimum.?

Question

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by C(x) = 2900x + 1,278,900/x , where x is the number of machines used.
Find the critical numbers of C(x).
Use the Second Derivative Test to determine the critical number that gives a local minimum.
What is the minimum average cost?

Answer 1

You find C'(x) = 2900 - (1 278 900)/x^2. The critical values are the values x such that C'(x) = 0, so x = 21 and -21 are critical.

The secomd derivative is C"(x) = 2*(1 278 900)/x^3. You find C"(21) > 0, so C(x) has a minimum at x = 21. Furthermore, C"(-21) < 0, so C has a max at x = -21.

Finally, to find the min avg cost, evaluate C(21).