I think the solution can be improved as the hint does not work in general (indeed points around the asymptote lead to conflicting answers). The previous question (not answered so far) asks us to find intervals where the function is increasing and decreasing which seems like natural information to use to find critical points.

Thanks for the feedback. However, for a differentiable function, a point in its domain is a critical point (local maximum or local minimum) iff the derivative at that point is zero, and the domain of this function excludes zero.