Problem about sum of polynomials

Question

I have this problem I don't know how to solve:

Let $f(x)$ be a polynomial of degree $n$ with real coefficients and such that $f(x) \geq 0 \forall x \in \mathbb{R}.$ How do I show that $f(x) + f'(x) + \cdots + f^{(n)}(x) \geq 0 \forall x \in \mathbb{R}?$ where $f^{(k)}(x)$ denotes the kth derivative of $f(x).$

Answer 1

For n=2, f(x) = ax$^2$ +bx +c, f(x) >=0 implies a>0, b$^2$ -4ac <=0, f(x) + f'(x) + f''(x) = ax$^2$ + (2a+b)x + ( c+ b +2a), (2a+b)$^2$ - 4a(2a+b +c) = b$^2$ -4ac -4a$^2$ <0, so f(x) + f'(x) + f''(x) > 0