Suppose $f(x)$ is two-times differentiable, $f(x),\;f'(x),\;f''(x)$ are all $>0$ and there are $a,\,b>0$ such that $f''(x)\leq af(x)+bf'(x)$

Question

Suppose $f(x)$ is two-times differentiable in $\mathbb R$, $f(x),\;f'(x),\;f''(x)$ are all $>0$ and there are $a,\,b>0$ such that $$ f''(x)\leq af(x)+bf'(x),\qquad\text{for all $x\in\mathbb R$}. $$ Show that

(1) $\displaystyle\lim_{x\to-\infty}f'(x)=0$.

(2) There is a constant $c$ such that $f'(x)\leq cf(x)$.

(3) Find the smallest $c$ in (2).

It's easy to prove (1), since otherwise $f(x)$ will not keep greater than zero.