So I am going through problems about derivatives, inflection points etc. Now When I am given a differentiable function, and I am supposed to find the interval when it increases/decreases. I found this neat trick to find that out. What I do is find all the zeroes of the derivative, then by Darboux's theorem between consecutive zeroes the function has to be either all positive or all negative, is that right? This saves me a lot of time since I do not have to draw the function itself. My question here, is this correct and does it always work, perhaps there are easier ways?
Also, regarding inflection points. By definition, it is a point, where the concavity changes, so where the second derivative changes the sign. Clearly, the necessary condition is that the inflection point is the critical point of $f'(x)$ but what are some other nice conditions that do not require me to know if the sign actually changes (to avoid drawing it). I think I have read that is is necessary but still not sufficient that all the derivatives have to equal to 0 at the point for it to be an inflection point.
I am trying to raise my proficiency/speed due to the nature of the GRE. Thank you!