Find the number of units x that produces the minimum average per unit C bar in the given equation

To find the minimum average cost number of units, we first have to find the average cost function. As the average cost is the total cost divided by the number of units produced, this will be:

AC=0.001x3+9x+2x=0.001x2+9+2x−1AC=0.001x3+9x+2x=0.001x2+9+2x−1

The number of units that minimize the AC will be found by equating the derivative of AC to zero and solving for x.

AC′=0.002x−2x−2AC′=0⇒0.002x−2x−2=0x=10AC′=0.002x−2x−2AC′=0⇒0.002x−2x−2=0x=10

Thus, the average cost is minimized at a production level of 10.