Question

How do you find the critical points of `f(x) = x - 3ln(x)`?

Answer 1

I use the definition: a critical point for a function, `f` is a point (number, value), `c` in the domain of `f` at which either `f'(c)=0` or `f'(c)` does not exist.

`f(x) = x-3lnx`

`f'(x)=1-3/x=(x-3)/x`

For this function, `f'(x)` does not exist for `x=0`, but the domain of `f` is `(0,oo)`, so `0` is not a critical point for `f`.

`f'(x)=0` at `x=3` which is in the domain of `f`, so `3` is a critical number for `f`.

(I understand that there are some who would make the critical point `(9, 3-3ln3)`.)