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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 Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
Solution
Let us consider that the company increases the annual subscription by ₹ 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
Hence, in order to get maximum profit, the company should increase its annual subscription by ₹ 100.
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