To begin, we are given the cost function and need to find the average cost function. Since the average cost of producing x items would be the total cost divided by the number of items produced, x, we can say:

[Math Processing Error]C¯(x)=C(x)x=x3+31x+432x=x2+31+432x−1

Now, if we are interested in finding the absolute minimum values of this average cost function on set intervals, we'll need to know if any critical numbers exists. So, let's take a derivative and find the critical numbers:

[Math Processing Error]C′¯(x)=2x−432x−20=2x−432x2432x2=2x2x3=432x3=216x=6.

Now, the derivative is undefined when x=0. However, the average cost function is also undefined there, and the value is not on either of the given intervals, so we'll ignore that. Now x=6 is a critical number that will play an important role.

(a) Let's look at the interval [Math Processing Error][1,10]. Note that the critical number of 6 lies on this interval. We need to check the endpoints of the interval as well as this critical number for the absolute min.

[Math Processing Error]C¯(1)=464C¯(6)=139C¯(10)=174.2.

From here we can conclude the minimum average cost on the given interval occurs when 6 items are produced and it is $139 per item.

(b) Now let's look at the interval [Math Processing Error][10,20]. Note that the critical number of 6 does not lie on this interval. We need only to check the endpoints of the interval.

[Math Processing Error]C¯(20)=452.6C¯(10)=174.2.

Therefore the absolute minimum average cost on this interval occurs when 10 items are produced and the average cost is $174.20 per item.