Question

A telephone company in a town has $500$ subscribers on its list and collects fixed charges of $Rs.300$ per subscriber. The company proposed to increase the annual subscription and it is believed that every increase of $R.1$ one subscribed will discontinue the service. Find what increase will bring maximum revenue?

Answer 1

Consider that company increases the annual subscription by $Rsx$
So $x$ subscribes will discontinue the service .
$\therefore$ Toral revenue of company after the increment is given by
$R\left(x\right)=(500-x)(300+x)$
$=15\times {10}^4+500-x-300x-x^2$
$=-x^2+200x+150000$
on differentiating both sides w.r.t. x we get
$R^{\prime }\left(x\right)=-2x+200$
Now $R^{\prime }\left(x\right)=0$
$\Rightarrow 2x=200\Rightarrow x=100$
$\therefore R"\left(x\right)=-2<0$
So $R\left(x\right)$ is maximum when $x=100$
hence the company should increase the subscription fee by $Rs100$ so that it has maximum profit.