# 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?

Feb 3, 2015

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 $x = - \frac{b}{2 a}$ (in your case $b = 5$ and $a = \frac{1}{9}$);
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:
$C ' = 5 + \frac{2}{9} x$
Set it $= 0$;
$5 + \frac{2}{9} x = 0$
$x = - \frac{45}{2} = - 22.5$
You can check using method 2).

And Graphically: