Find the minimum value of the average cost for the given cost function on the given intervals

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.