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?

Dear Student

Let
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

= -x^2+ 200x + 150000
Differentiating both sides w.r.t. x,
R’(x) = -2x + 200

For local maxima and local minima,
R’(x) = 0 -2x + 200 = 0
⇒x = 100

R’’(x) = -2 < 0
Thus, R(x) is maximum at x = 100
Thus, in order to get maximum profit, the company should increase its annual subscription by Rs100

Thanks