A real polynomial $P$ of degree $n$ in one real variable is hyperbolic if its roots are all real. A real-valued function $P$ is called a hyperbolic polynomial-like function of degree $n$ if it has $n$ real zeros and $P^{(n)}$ vanishes nowhere. Denote by $x_k^{(i)}$ the roots of $P^{(i)}$, $k=1,\dots,n-i$, $i=0,\dots,n-1$. Then, in the absence of any equality of the form

\begin{equation} x_i^{(j)}=x_k^{(l)} \tag{*} \label{*} \end{equation}

for all $i<j$ we have

\begin{equation} x_k^{(i)}<x_k^{(j)}<x_{k+j-i}^{(i)} \tag{**} \label{**} \end{equation}

(the Rolle theorem). For $n\geq4$ (respectively, for $n\geq5$) not all arrangements without equalities \eqref{*} of $\tfrac12n(n+1)$ real numbers $x_k^{(i)}$ and compatible with \eqref{**} are realizable by the roots of hyperbolic polynomials (respectively, of hyperbolic polynomial-like functions) of degree $n$ and of their derivatives. For $n=5$ and when

$$ x_1^{(1)}<x_1^{(3)}<x_2^{(1)}<x_3^{(1)}< x_2^{(3)}<x_4^{(1)} $$

we show that all such 102 arrangements are realizable by hyperbolic polynomial-like functions (of which 66 are obtained by hyperbolic polynomials and another 8 by perturbations of such).