TITLE: Prove that If $f$ is polynomial function of even degree $n$ with always $f\geq0$ then $f+f'+f''+\cdots+f^{(n)}\geq 0$.
QUESTION [25 upvotes]: I can't solve this problem:
Suppose $f$ is polynomial function of even degree $n$ with always $f\geq0$.
Prove that
$f+f'+f''+\cdots+f^{(n)}\geq 0$.

REPLY [46 votes]: Alternatively, by a standard Maximum Value Theorem argument, the even-degree polynomial $g=f+f'+\dots+f^{(n)}$ has a global minimum at some point $a$. Then $g(a)=f(a)+g'(a)\ge 0$, so $g\ge g(a)\ge 0$.