There exist a function $f(x)$ satisfying $f(0)=1$, $f'(0)=-1$, $f(x) > 0$ for all $x$

Question

$\begin{array}{1 1}(a)\;f''(x)> 0\;for\;all\;x\\(b)\;-1 < f''(x) < 0\;for\;all\;x\\(c)\;-2\leq f''(x)\leq -1\;for\;all\\(d)\;f''(x)< -2\;for\;all\;x\end{array}$