Question

A telephone company in town has 500 subscribers on its list and collects fixed charges of 300 per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of 1, one subscriber will discontinue the service. Find what increase will bring maximum profit.

Answer 1

Consider the company increases the annual subsrciption by x.
So, x subscribers will discontinue the service.
$\therefore$ Total revenue of company after the increment is given by
$R\left(x\right)=(500-x)(300+x)=15\times {10}^4+500x-300x-x^2=-x^2+200x+150000$
On differentiating both sides w.r.t. x, we get

$R^{\prime }\left(x\right)=-2x+200Now,R^{\prime }\left(x\right)=0\Rightarrow 2x=200\Rightarrow x=10\therefore R^{\prime \prime }\left(x\right)=-2<0$
So, R (x) is maximum when x = 100.