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...