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

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Answer 1 · 2015-12-03T17:02:35

That function has no critical points.

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.

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