Let $f(x)=xe-x.$

$($a$)$ Find the critical numbers for $f.$

$($b$)$ Determine the intervals where $f$ in increasing or decreasing.

$($c$)$ Find the intervals where $f$ is concave up or concave down.

$($d$)$ Find the points of inflection for $f.($Warning: review the definition of inflection point; it might not be what you think it is$!)$

$($e$)$ Using $(1)-(4),$ and plotting a few points, draw the graph of $f(x)=xe-x.$