Is there a term for a smooth $\mathbb R\to\mathbb R^+$ function that is [unimodal](https://en.wikipedia.org/wiki/Unimodality#Unimodal_function) (has one local maximum), and whose $n^{\text{th}}$ derivative has $(n+1)$ local extrema? An example of such a function is the Gaussian function $f(x)=\exp(-x^2)$.

[![Plot of the Gaussian function and its derivatives](https://i.stack.imgur.com/UiZCG.png)](https://i.stack.imgur.com/UiZCG.png)

A negative example of a unimodal function that does not fit these criteria is $f(x)=1/(10+x^4)$. Its derivatives have too many extrema.

[![Plot of the rational function and its derivatives](https://i.stack.imgur.com/dEKLr.png)](https://i.stack.imgur.com/dEKLr.png)

Another negative example that shows that the number of extrema can grow exponentially (with respect to the order of a derivative) is the Rvachev function $f(x)=\operatorname{up}\left(x/4\right)$.

[![Plot of the Rvachev function and its derivatives](https://i.stack.imgur.com/M2O1G.png)](https://i.stack.imgur.com/M2O1G.png)