Compare growth of function and derivative

Question

Suppose $f:\mathbb R \rightarrow \mathbb R$, $O(1)<f(x)<O(x)$ (i.e. has slower than linear growth but is unbounded), $\lim_{x\rightarrow \infty} f'(x)=0$ and $f'$ is monotonic. Is it possible to show that there exists a $\delta$ such that for $|x|>\delta$, $|x f'(x) |< |f(x)|$ for all such functions $f$?