It is quite easy to see that there are plenty of functions $f$ for which
$$ \frac{d^n}{dx^n} f(x) \geq 0 \;\;\;\forall n \in \mathbb{N}, x \in \mathbb{R}_+$$
For starters, it holds for any polynomial with positive or zero coefficients. The stronger condition, with a strict inequality
$$(1) \;\;\;\frac{d^n}{dx^n} f(x) > 0 \;\;\;\forall n \in \mathbb{N}, x \in \mathbb{R}_+$$
holds in less cases. However, sums of exponentials of the form
$$ \;\;\; f(x) = \sum_i A_i e^{B_i x} \;\;\; A_i, B_i > 0$$
satisfy this stronger condition. But there are others
My questions are:
1) Does (1) imply $O(f(x)) \geq O(\exp(x))$,
(in the sense $O(a(x)) >,=,< O(b(x))$ if $\lim_{x\rightarrow\infty} \frac{a(x)}{b(x)} = \infty,\text{constant},0$, respectively)
2) Is there a simple test for (1), and/or, does this condition have an established name?