Suppose that the average cost, in dollars, of producing a shipment of a certain product is C = 5,000x + 20,000/x, x > 0 where x is the number of machines used in the production process. (a) Find the critical values of this function. (Assume 0 < x < [infinity]. Enter your answers as a comma-separated list.) x = Incorrect: Your answer is incorrect. (b) Over what interval does the average cost decrease? (Enter your answer using interval notation.) (c) Over what interval does the average cost increase? (Enter your answer using interval notation.)

The average cost function is C(x) = 5000x + 20,000/x, where x is the number of machines used in the production. The critical value is C(x) = 20,000 and it happens when x = 2.

If we have a function f(x), the critical point happens when its first derivative is equal to zero.

f '(x) = 0

In the given problem, the function is:

C(x) = 5000x + 20,000/x

Take the derivative:

C '(x) = 5000 - 20,000/x² = 0

5000 x² = 20,000

x² = 4

x = ±2

Since x is within the interval: 0<x<∞, the solution is x = 2

Substitute x = 2 into the function:

C(2) = 5000 (2) + 20000/2

C(2) = 10000 + 10000 = 20,000

Hence, the critical value is C(x) = 20,000 and it happens when x = 2.

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