Before we can analyze the average cost function, we first need to define it. We can do so by dividing each term in the cost function by the quantity being sold, which is our variable, x.

¯C(x)=C(x)x=0.06x3−6.0x2+171xx=0.06x2−6x+171C¯(x)=C(x)x=0.06x3−6.0x2+171xx=0.06x2−6x+171

Now that we have the average cost function, we can find the quantity that minimizes it. This will first require us to differentiate this function, which will only need the Power Rule as a technique.

¯C′(x)=0.12x−6C¯′(x)=0.12x−6

Setting this equal to zero will reveal the critical point of the average cost function.

0.12x−6=00.12x=6x=500.12x−6=00.12x=6x=50

It's thus possible that producing 50 wheels minimizes the daily average cost. However, we need to confirm this by using the second derivative test.

¯C′′(x)=0.12C¯″(x)=0.12

Since the second derivative is positive at a critical point, this quantity we found is indeed the quantity to produce to minimize the average cost. We can find this average cost by evaluating the average cost function at this production level.

¯C(50)=0.06(50)2−6(50)+171=21C¯(50)=0.06(50)2−6(50)+171=21

Thus, the minimum average cost of product is $21 per wheel, which occurs when 50 wheels are produced daily.