Polynomial or Exponential PROBLEM:
Let $f(x)$ be a polynomial function.
It is known that for every $x$:
$$ f'(x) \leq f(x) $$
Prove/disprove:
For every $x$:
$$ f(x) \geq 0 $$
MY INTUITION:
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?

Assume that $f<0$ somewhere. There are only two cases: (i) $f$ attains a critical point with a strictly negative value. But then, if $x^*$ is this point, $f(x^*) \geq f'(x^*) =0$, a contradiction. The other case is that $f<0$ definitely. As you noticed, considering either $f(x)e^{-x}$ or $f(x)e^x$, you can prove that $f$ goes to $-\infty$ exponentially fast, and this is impossible since $f$ is a polynomial.
I think this approach can be made completely rigorous.
