Question

How do you find the value of x that gives the minimum average cost, if the cost of producing x units of a certain product is given by `C = 10,000 + 5x + (1/9)x^2`?

Answer 1

Your function is a quadratic and can be represented by a PARABOLA (basically the shape of an "U" upwards or downwards oriented).

Your parabola is in the shape of "U" because the coeficient of `x^2` is >0.

You basically are looking for a minimum (the vertex) of your parabola (the bottom of your "U" shape) and the corresponding `x` value.

To find this you have various methods:

1) plot your function and look for it;
2) using the fact that the vertex has coordinate `x=-b/(2a)` (in your case `b=5` and `a=1/9`);
3) Derive your function and set the derivative equal to zero (this gives you the point of inclination equal to zero or your minimum).

I would use the derivative stuff:
`C'=5+2/9x`
Set it `=0`;
`5+2/9x=0`
`x=-45/2=-22.5`
You can check using method 2).

And Graphically: