Finding the Minimum Average Cost

Date: 2/12/96 at 12:27:1
From: Anonymous
Subject: average cost / calculus

I've had problems with this one.

"The cost of producing x units of a certain product 
is given by C=10,000 + 5x + (1/9)x^2.    

Find the value of x that gives the minimum average cost."

The problem also listed these following multiple choice answers:

a)  30,000
b)  300
c)  3,000
d)  30
e)  None of these

Now, we have the correct answer, what we need is the actual way to do 
this problem.

                  Thank You, 

                  Love, Erin

Date: 2/13/96 at 18:35:11
From: Doctor Ethan
Subject: Re: average cost / calculus

Hi there.  I think I have the method you're looking for.

You have a cost function for x units given by:

C = 10,000 + 5x + (1/9)x^2

Now, the cost per unit is going to be that same function divided by x.
Let's call this function U:

U = 10,000/x + 5 + (1/9)x

To find the minimum total cost, you need to find the minimum of 
this function.  Analyzing the derivatives should get you the answer 
you need:

U'  = -10,000/(x^2) + 1/9
U'' = 20,000/(x^3)

(You might want to try this yourself before reading on...)

By examining U'= 0, we find that x = 300 is a critical point.
Further, since U'' is positive, the function must be at a minimum at 
that point.

I sure hope I came up with the answer you were looking for.

-Doctor Ethan,  The Math Forum