For me, it is a folklore result for people interested in Descartes' rule of signs, even though I am not aware of any paper stating it explicitly. It follows from Descartes' rule of signs.

**Theorem** (Descartes' rule of signs). The number of positive roots of a real polynomial is not larger than the number of sign changes in the sequence of its coefficients.

You can prove the following generalization: 

**Lemma.** Let $P$ be a degree-$d$ polynomial and $r\in\mathbb R$. The number of sign changes in the sequence of the $P^{(n)}(r)$ for $n=0$ to $d$ is an upper bound on the number of roots of P larger than $r$. 

*Proof.* Let $Q(x)=P(x+r)$. Then $Q^{(n)}(0)/n!$ is the coefficient of $x^n$ in $Q$. Therefore, the number of signs changes in the sequence $(P^{(n)}(r))_n$ equals the number of sign changes in the sequence of coefficients of $Q$. By Descartes' rule of signs, this number of sign changes upper bounds the number of positive roots of $Q$. $\square$ 

*Remark.* It is a consequence of Rolle's Theorem since Rolle's Theorem is used to prove Descartes' rule of signs.