Solutions

Let

$$g = f + f' + f'' + \cdots + f^{(n)}.$$

Then

$$g' = f' + f'' + \cdots + f^{(n)}$$

and

$$f = g - g' \ge 0.$$

Suppose for contradiction that there exists $c$ such that $g(c) < 0$. Notice that $g$ is also a polynomial of even degree $n$ (since $f$ features in its sum) so

$$\lim_{x \to -\infty} g(x) = \lim_{x \to \infty} g(x) = \infty$$

and therefore by the intermediate value theorem there exist $u$ and $v$ such that

$$g(u) = g(v) = 0,$$

$u < c < v$ and $g(x) < 0$ for $x \in (u,v)$. By Rolle's theorem, there exists $y \in (u,v)$ such that $g'(y) = 0$. But this implies that

$$f(y) = g(y) - g'(y) = g(y) < 0,$$

a contradiction.