TITLE: Prove $u\left( x\right)=W\left( x\right)+W'\left( x\right)+W''\left( x\right)+ \cdots \ge 0.$
QUESTION [5 upvotes]: Let $W\left( x\right) \ge 0$ for $x \in \mathbb{R}$ be a polynomial. 
Prove $$u\left( x\right)=W\left( x\right)+W'\left( x\right)+W''\left( x\right)+ \cdots \ge 0.$$ Is there a simple way?

REPLY [4 votes]: Suppose for a contradiction $u(x_0)<0$ for some point $x_0\in \mathbb{R}$. Then, because $u$ is a polynomial of even degree with positive leading coefficient (because $W$ must have these properties as well), there must be an interval $[a,b]\ni x_0$ such that $u(x)<0$ for all $x\in ]a,b[$ and $u(a)=0=u(b)$. Then there must be some $c\in ]a,b[$ such that $u'(c)=0$ by Rolle's theorem. Hence
$$0=u'(c)=W'(c)+W''(c)+\dots =u(c)-W(c),$$and thus
$$0\leq W(c)=u(c)<0,$$
which is a contradiction.