A few riddles behind Rolle's theorem

First year undergraduates usually learn about classical Rolle'stheorem which says that between two consecutive zeros of a smooth univariate function $f$, one can always find at least one zeroof its derivative $f^\prime$. In this paper, we study a generalization of Rolle's theorem dealing with zeros of higherderivatives of smooth univariate functions enjoying a natural additional property. Namely, we call a smooth function whose $n$th derivative does not vanish on some interval $I\subseteq \bR$ a {\it polynomial-like function of degree $n$ on $I$}. We conjecture that for polynomial-like functions of degree $n$ with $n$ real distinct roots there exists a non-trivial system of inequalities completely describing the set of possible locations of their zeros together with their derivatives of order up to $n-1$. We describe the corresponding system of inequalities in the simplest non-trivial case $n=3$