# A company earns a weekly profit of p dollars by selling x items according to the function p = -x^2 + 125x - 2200. Find the number of items the company must sell each week to obtain the largest possible profit Then find the largest possible profit

A company earns a weekly profit of p dollars by selling x items according to the function $p=-{x}^{2}+125x-2200$. Find the number of items the company must sell each week to obtain the largest possible profit Then find the largest possible profit.
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kuglatid4
$p=-{x}^{2}+125x-2200\phantom{\rule{0ex}{0ex}}\frac{dp}{dx}=-2x+125\phantom{\rule{0ex}{0ex}}\frac{d{p}^{2}}{d{x}^{2}}=-2$
for maximum profit $\frac{dp}{dx}=0$ and $\frac{{d}^{2}p}{d{x}^{2}}<0$
$\frac{dp}{dx}=0⇒-2x+125=0\phantom{\rule{0ex}{0ex}}x=62\cdot 5$
Hence $x=62\cdot 5$ give maximum profit.
$P\left(62\cdot 5\right)=-\left(62\cdot 5{\right)}^{2}+125×62\cdot 5-2200\phantom{\rule{0ex}{0ex}}=1706.25$