Question

What are the critical points of `f(x) =(x^2-2x)/(x-1)^2`?

Answer 1

That function has no critical points.

Explanation:

Note that the domain of `f` is `(-oo,1) uu (1,oo)`

`f'(x) = ((2x-2)(x-1)^2-(x^2-2x)(2(x-1)))/(x-1)^4`

`= ((2x-2)(x-1)-(x^2-2x)(2))/(x-1)^3`

`= 2/(x-1)^3`

`f'(x)` is never `0` and

it is undefined at `x=1`, but `1` is not in the domain of `f`, so it is not a critical number.