I have this question
Find the critical points of $f(x) = \ln(x^3-3x+1)$.
This is a checkpoint question from my teacher and I don't understand the solution. Question and teacher solution here.
My first question is the point $x=+1$ is not defined in the parent function but I don't understand how to take the domain of this function I am getting stuck after this. \begin{align} x^3-3x+1 &> 0 \\ x^3-3 &>-1 \end{align}
I don't know how to proceed further.
Then to find critical points I know you have to find the first derivative and set it to $0$ and solve for $x$. I get $$ f'(x) = \frac{3x^2-3}{x^3-3x+1} $$ What I am stuck with is how to set the numerator equal to $0$ and solve for $x$ I get stuck there too.
Your teacher is not attempting to establish what the domain of the function is - they are merely checking whether the critical values found can be input into the function to give a real answer.
As far as finding the critical values are concerned, when you solve a fraction equal to zero you only need to solve the numeratore equal to zero.