# Lynbrook West , an apartment has 100 two-bedroom units. The montly profit realized from renting out x apartments is given by the following function

Lynbrook West , an apartment complex , has 100 two-bedroom units. The montly profit (in dollars) realized from renting out x apartments is given by the following function.
$P\left(x\right)=-12{x}^{2}+2136x-41000$
To maximize the monthly rental profit , how many units should be rented out?
What is the maximum monthly profit realizable?
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Step 1
Given profit function ,
$P\left(x\right)=-12{x}^{2}+2136x-41000$
To maximize profit
$\frac{d}{dx}P\left(x\right)=0$
$⇒\frac{d}{dx}\left(-12{x}^{2}+2136x-41000\right)=0$
$⇒-24x+2136=0$
$⇒-24x=-2136$
$⇒x=\frac{-2136}{-24}$
$x=89$
Here , $\frac{{d}^{2}P\left(x\right)}{{dx}^{2}}=-24<0$
as per second derirative test profit is maximum at $x=89$
Also , maximum profit
$=P\left(89\right)=-12{\left(89\right)}^{2}+2136\left(89\right)-41000$
$=54052$
To maximize the montly rental profit, 89 units should be rented out, 54052\$ be the maximum monthly profit