In some cases it turns out that,
If $f$ is strictly increasing, and convex then all its higher order derivatives are also strictly increasing and convex (assuming $f$ is continuously differentiable).
For example,
- $e^{a x}$ on $\mathbb{R}$ for $a>0$,
- $\tan(x)$ on $(0,\frac{\pi}{2})$ or,
- $x^n$ on $(0,\infty)$ - at least for the first few derivatives.
Of course several counter examples, eg. $x \log x$, which is convex and increasing on $(1,\infty)$ but its derivatives do not share this property.
So my question is that under what conditions does it happen that if $f$ is increasing then its derivatives are also monotone (when are they increasing or decreasing)??
By conditions I mean conditions of $f$, not its higher derivatives. I know this is somewhat unusual in that I am trying to infer information about f' knowing information about $f$. In most calculus courses the reverse is deduced.
Thanks for reading