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

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

$f(x)=e^{-x}$ is one such function.
Here $f(0)=1,f'(0)=-1$
$f(x) >0 \forall x$
$f''(x) >0 \forall x$
Hence (a) is the correct answer.
answered Dec 20, 2013