I was wondering about the analysis of derivatives in the study of a function. We know that
$f'(x) = 0$ can give us the critical points $x_k$ (maximum, minimum and sometimes inflection points), not always but it's not the point.
$f'(x) > 0$ lead us to find the intervals in which the function is increasing or decreasing.
$f''(x_k)$ can be useful to classify the critical points as maximum, minimum or inflection points.
$f''(x) = 0$ gives us the inflection points.
$f''(x) > 0$ lead us to find the interval in which the function is concave or convex.
Now my question is: is there some useful/interesting reason to study the derivatives beyond the second one?
I mean what could $$f'''(x) = 0$$ give us? What kind of points, if any?
And what $$f'''(x) > 0$$ could give us in terms of intervals?
(Clearly my question is related to those functions that can admit/have a third, or higher, derivative).
