# How do you find the critical numbers of y = (x^2-4)/(x^2-2x)?

Mar 23, 2017

First things first, the function can be simplified as follows:

$y = \frac{\left(x + 2\right) \left(x - 2\right)}{x \left(x - 2\right)}$

$y = \frac{x + 2}{x}$

$y = \frac{x}{x} + \frac{2}{x}$

$y = 1 + \frac{2}{x}$

The derivative of this is

$y ' = - \frac{2}{x} ^ 2$

There are critical numbers when the derivative is undefined or the derivative equals $0$. When the derivative is undefined at $x = 0$, the function is also undefined, so this is not a critical value.

$0 = - \frac{2}{x} ^ 2$

$0 = - 2$

This is obviously a contradiction, therefore the equation $- \frac{2}{x} ^ 2 = 0$ has no solution. This also means the given function has no critical values.

Hopefully this helps!