For this problem, consider the function, f$($x$)=($x$)($e$)^(($ x $^2)/2)$

$($a$)$ Find all the critical numbers of f$($x$).$

$($b$)$ Determine the intervals over which the function is increasing$/$decreasing$.$

$($c$)$ Classify each critical point as a local max$/$local min.

$($d$)$ Determine the intervals over which the function is concave up$/$concave down.

$($e$)$ Find all the inflection points of f$($x$).$