In normal calculus without stochastic processes, we look at $\frac{df}{dx} = 0$ to calculate critical points and $\frac{d^2f}{dx^2} = 0$ to calculate inflection points. For example, given function
$$f(x)=x^3-x$$
We can derive the critical points: $$f(x)=3x^2-1=0 \implies x=-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$$
And we can derive the inflection points: $$f(x)=6x=0 \implies x=0$$
What is the equivalent of a critical point and inflection point in ito calculus? What additional information do we need about the error term? For example, say we have the function
$$f(x)=x^3-x + \epsilon$$
How would you calculate the critical points and inflection points analytically? What about empirically if we did not know this specific function was what we were looking at?
Edit: Revised incorrect $x=1,0,-1$ critical points.