The profit function $P\left(x\right)$ of a company; selling $x$ items per day is given by $P\left(x\right)=(150–x)x–1600$. Find the number of items that the company should sell to get maximum profit. Also find the maximum profit.

Given profit function 

$P\left(x\right)=(150-x)x-1600=150x-x^2-1600$

Let $f\left(x\right)=150x-x^2-1600$

$f^{\mathrm{\prime }}\left(x\right)=150-2x\Rightarrow f^{\prime \prime }\left(x\right)=-2<0(max)$

Clearly it has max profit

Consider $f^{\mathrm{\prime }}\left(x\right)=0 ;$

$150-2x=0;x=75$

the company should sell 75 items to get max profit

$P\left(x\right)=(150-75)75-1600$

$=\left(75\right)75-1600=4025$