A $12-$in$.-$by$-36-$in$.$ sheet of cardboard is folded in half to form a $12-$in$.-$by$-18-$in$.$ rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid. Complete parts $($a$)$ through $($f$)$ below.

c$.$ Use a graphical method to find the maximum volume and the value of $x$ that gives it$.$

The maximum volume is

$($Type a whole number.$)$

The maximum volume occurs for $x=$

$($Round to one decimal place as needed.$)$

d$.$ Confirm your result in part $($c$)$ analytically.

The maximum volume occurs at exactly $x=,$ which confirms the result from part $($c$).$

$($Type an exact answer, using radicals as needed.$)$

e$.$ Find a value of $x$ that yields a volume of $448$ in$.{?}^3$

$x=$

$($Type an exact answer, using radicals as needed.$)$

f$.$ Which of the following is an issue that arises in part $($b$)?$

A$.$ The domain is restricted to $x-$values that cause $V(x)$ to be positive and that cause each dimension of the box to be positive.

B$.$ The domain is restricted to $x-$values that cause at least one dimension of the box to be positive.

C$.$ The domain is only restricted to $x-$values that cause $V(x)$ to be positive.

D$.$ There are $x-$values that cause $V(x)$ to be undefined.