A telephone company in a 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 per one subscriber will discontinue the service. Find what increase will bring maximum profit?

₹100

The correct answer is Option (2) → ₹100 ##

Consider that company increases the annual subscription by ₹ $x$.

So, $x$ subscribers will discontinue the service.

$∴$ Total revenue of company after the increment is given by

$R(x) = (500 - x)(300 + x) = 15 \times 10^4 + 500x - 300x - x^2$

$= -x^2 + 200x + 150000$

On differentiating both sides w.r.t. $x$, we get

$R'(x) = -2x + 200$

Now, $R'(x) = 0 \Rightarrow 2x = 200 \Rightarrow x = 100$

$∴R''(x) = -2 < 0$

So, $R(x)$ is maximum when $x = 100$.

Hence, the company should increase the subscription fee by ₹ 100, so that it has maximum profit.