`Does anyone have any advice or help on how to tackle the following problem` ?: 
$$
\mbox{Show that the function}\
\operatorname{f}\left(r\right) =
\int_{0}^{\infty}\operatorname{J}_{0}\left(rx\right)
{\rm e}^{-ax^{2}}\cos\left(x\right)x\,{\rm d}x\,,\ r \geq 0
$$
`is positive` for sufficiently large $a$.
 - It is easy to check that $\operatorname{f}\left(0\right)$ is positive for sufficiently large a and
$\operatorname{f}\left(r\right) \to 0$ as $r \to \infty$.
 - So if I can prove that f cannot have any zeros (or local minima) it would prevent it from becoming negative.
 - I am not sure though if looking at the critical points of $\operatorname{f}$ is the right way to go.

Thanks in advance for any help :) I also, even help such as gaging how easy/difficult such a question is welcome.