Question

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

Answer 1

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

`y = ((x + 2)(x - 2))/(x(x - 2))`

`y = (x+ 2)/x`

`y = x/x + 2/x`

`y = 1 + 2/x`

The derivative of this is

`y' = -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 = -2/x^2`

`0 = -2`

This is obviously a contradiction, therefore the equation `-2/x^2 = 0` has no solution. This also means the given function has no critical values.

Hopefully this helps!