units.

Find the revenue function, $R(x).$

$R(x)=$

Find the cost function, $C(x).$

$C(x)=$

Find the profit function, $P(x).$

$P(x)=$

$($a$)$ Find $P^{\text{'}}(x).$

$P^{\text{'}}(x)=$

Considering the limitations of production, find the quantity $($in hundreds of units$)$ that will give the maximum profit. hundred units

$($b$)$ Find the maximum profit. $($Round your answer to the nearest cent.$)$