$2.$ Now consider a rectangular piece of paper with length $\backslash ($ a $\backslash )$ and width $\backslash ($ b $\backslash ),$ with $\backslash ($ a$>$b $\backslash ).$ Again imagine cutting squares of length $\backslash ($ x $\backslash )$ units out of each of the four corners.

$($a$.)$ Draw a labeled sketch of this piece of paper, indicating $\backslash ($ a$,$ b $\backslash ),$ and $\backslash ($ x $\backslash )$ on your sketch. Also draw a labeled sketch of the open$-$top box.

$($b$.)$ Write down a function for the volume $\backslash ($ V$($x$)\backslash )$ of the open$-$top box formed by folding up the sides.

$($c$.)$ At what real$-$world value of $\backslash ($ x $\backslash )$ is the volume maximized? $($There will be two critical points for $\backslash ($ V$($x$)\backslash )$; show that one of these is not physically possible.$)$

$($d$.)$ Draw by hand a sketch of $\backslash ($ V$($x$)\backslash )$ as a function of $\backslash ($ x $\backslash ).$ Label your sketch to show the $\backslash ($ x $\backslash )-$value yielding maximum volume.