Question

What are the critical points of `f(x) =(x^2-4x+5)/(x-3)`?

Answer 1

The critical numbers are `1` and `7`.

Explanation:

`f'(x) = ((2x-4)(x-3)-(x^2-4x+5)(1))/(x-3)^2`

`= (x^2-6x+7)/(x-3)^2`

`f'` fails to exists only at `x=3` which is not in the domain of `f`,

and `f'(x) = 0` at `x=1` and at `x=7`, both of which are in the domain of `f`.

So, the critical numbers for `f` are `1` and `7`.

There appears to be some variability in the use of the term "critical point".
I use it to mean a point in the domain of a function at which the derivative is `0` or fails to exist.
Some people seem to use it to mean a point on the graph of a function at which the derivative is `0` or fails to exist.