Let $f(x)$ be a polynomial function.
It is known that for every $x$:
$$ f'(x) \leq f(x) $$
For every $x$:
$$ f(x) \geq 0 $$
Suppose by contradiction that $f(z)<0$ for some $z$. Then $f'(z)<0$ too, so $f$ must go down and down to $-\infty$.
Moreover, the rate of decrease must by at least exponential because:
$$ |f'(x) \geq f(x)| $$
and we know that equality holds for the exponential function.
This contradict the fact that $f(x)$ is polynomial.
MY QUESTIONS: Is my intuition true? If so, how to formalize it? If not, then what is the correct answer?