Math Help - finding absolute maximum and absolute minimum

1. finding absolute maximum and absolute minimum

I need to find the absolute maximum and minimum of the function below on the interval [0,4]
g(x) = x / x^2 + 4

I know that i need to find g' which i think is
g'(x) = (x^2 + 4)1 - x(2x)
(x^2 + 4)^2
g'(x) = 4 - x^2
(x^2 - 4)^2

Am I correct so far? Do I need to do g'' or should I factor out and find the critical values?

2. I get

$g(x) = \frac{x}{x^2+4}$

$g'(x) = \frac{(x^2+4)\times 1 - 2x \times x}{(x^2+4)^2}$

$g'(x) = \frac{x^2+4 - 2x^2}{(x^2+4)^2}$

$g'(x) = \frac{4 - x^2}{(x^2+4)^2}$

making g'(x)=0

$0 = \frac{4 - x^2}{(x^2+4)^2}$

$0 = 4 - x^2$

$x=\pm 2$

You can sub these values into g''(x) if you want to know if they are mins or maxs. Or if you know the general shape of the orignal function it shouldn't be required.