Finding minimum value of function

[rozzali]

Hi,

I cannot figure out how this person arrived at the equation which determined the minimal value function. Here is what I have;

The total cost of producing x units of a certain product is given by;
C(x)=1/5x^2+2x+2000

The average cost for producing x units is
C1(x)=C(x)/x

A. Find the rational function C1(x); (I understand this it is simple)
(1/5x^2+2x+2000)/x = 1/5x+2+2000/x

B. At which production level will the average cost per unit be minimal? (THIS IS WHERE I CANNOT UNDERSTAND HOW THEY GOT THEIR ANSWER;
The minimal value function C1=(c/a)^1/2 where c=2000 and a=1/5 (how do you arrive at this equation) I have the answer but I need to understand how I can get this equation. I realize the value of c and a are from the original equation but I do not know why. Please Help!!!

 

[galactus]

Note that the minimum value given is \(\displaystyle \sqrt{\frac{2000}{\frac{1}{5}}}=100\)

If we differentiate C1, we get \(\displaystyle \frac{-2000}{x^{2}}+\frac{1}{5}\)

Set this to 0 and solve for x results in x=100, as your formula gives.