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?
Answer & explanation
Correct answer: option 2
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.