We've been given the cost function. We can find the marginal cost function as it is the first derivative of the cost function.

C(x)=500x−0.005x2+10−8x3C′(x)=MC(x)=500−0.01x+3∗10−8x2C(x)=500x−0.005x2+10−8x3C′(x)=MC(x)=500−0.01x+3∗10−8x2

The marginal cost at x=6000 will be:

MC(6000)=500−0.01∗6000+3∗10−8∗60002=441.08MC(6000)=500−0.01∗6000+3∗10−8∗60002=441.08

The actual cost C(6001) - C(6000) will be:

C(6001)−C(6000)=500∗6001−0.005∗60012+10−8∗60013−(500∗6000−0.005∗60002+10−8∗60003)=441.075C(6001)−C(6000)=500∗6001−0.005∗60012+10−8∗60013−(500∗6000−0.005∗60002+10−8∗60003)=441.075

The average cost is the total cost divided by the number of units produced. Therefore, the average cost for 6000 units will be:

AC=500∗6000−0.005∗60002+10−8∗600036000