Suppose the total cost function for manufacturing a certain product is C(x)=0

We are given the cost function for manufacturing a certain product to be:

C(x)=0.002x2+24C(x)=0.002x2+24

where x represents the number of units of the product that are produced. Differentiating the cost function gives the marginal cost as follows:

C′(x)=0.004xC′(x)=0.004x

Dividing the cost function by the production level x gives the average cost:

¯C(x)=0.002x2+24x=0.002x+24xC¯(x)=0.002x2+24x=0.002x+24x

The average cost is minimized when the marginal cost equals the average cost as follows:

0.004x=0.002x+24x0.002x=24xx2=240.002x2=12,000x≈109.54450.004x=0.002x+24x0.002x=24xx2=240.002x2=12,000x≈109.5445

We can neglect the negative root to the above equation because units are a positive quantity. Therefore, the average cost is minimized when 110 units are produced rounded to the nearest whole number of units.