Consider a rectangle formed under the line $y=8-\frac{4}{3}x$ in the first quadrant, where $x$ is in $0,6.$ Include the following in your solution.

$($a$)$ A formula that gives the area of the rectangle in terms of $x.$

$($b$)$ Work determining the critical number$($s$)$ of your function.

$($c$)$ What the maximum obtainable area is$.$

$($d$)$ An explanation using the extreme value theorem or other concepts for why this value is indeed a maximum.