If $c_{n} > 0$ then $\sum_{0}^{n}c_{k}x^{k} > 0$ for some $x \in \mathbb{R}$? Let $n \geq 1$ be an integer and let $c_{0}, \dots, c_{n} \in \mathbb{R}$. If $c_{n} > 0,$ is there necessarily an $x \in \mathbb{R}$ such that 
$$\sum_{0}^{n}c_{k}x^{k} > 0?$$
I just realized that for a while I had implicitly taken this for granted. However, when I would like to give a rigorous proof then I find it is not that obvious. 

Yes.
Suppose $x>0$ and consider $f(x) = {\sum_{k=0}^n c_k x^k \over x^n} = \sum_{k=0}^n c_k {1 \over x^{n-k}}$. Then $\lim_{x \to \infty} f(x) = c_n>0$, hence there is some $M$ such that if $x \ge M$, then $f(x) >0$. Hence
${\sum_{k=0}^n c_k x^k } >0$ for $x \ge M$.
