**Question:**

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

**Solution:**

Let’s consider that the company increases the annual subscription by Rs x.

So, x is the number of subscribers who discontinue the services.

Total revenue, R(x) = (500 – x) (300 + x)

= 150000 + 500x – 300x – x2

= -x2 + 200x + 150000

Differentiating both sides w.r.t. x, we get R’(x) = -2x + 200

For local maxima and local minima, R’(x) = 0

-2x + 200 = 0 ⇒ x = 100

R’’(x) = -2 < 0 Maxima

So, R(x) is maximum at x = 100

Therefore, in order to get maximum profit, the company should increase its annual subscription by Rs 100.