Greetings,


I am currently working on a paper that involves an upper bound of the largest root of a polynomial. With the help of the Mean Value Theorem, I believe a colleague and I have proved the following: if f^(n)(p) > 0 (the n-th derivative evaluated at "p", for n=0,1,2..d, where d is the degree of the polynomial), then p>r, where r is the largest root of the polynomial.


The only question I have is the following: is this well-known or has been cited/proven elsewhere? It just seems basic enough that someone in the past has referenced it in a paper/Lemma. Though we could be mistaken and have a flaw in our own proof...


thanks,


Mike



A known Lemma on the largest root of a polynomial and its derivatives?
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.


