An internet service provider in a town has 500 subscribers and charging Rs. 300 per month from each subscriber. He proposes to increase the monthly charges and it is believed from past experience that for every increase of Rs. 1, one subscriber will discontinue the service.
The number of subscribers for which the maximum revenue is earned is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 400
With an increase of Rs. x, the charge is Rs. (300 + x) and 500 − x subscribers remain:
\(R(x) = (300 + x)(500 - x)\)
\(R(x) = 150000 + 200x - x^2\)
For a maximum, R’(x) = 0:
\(R'(x) = 200 - 2x = 0\)
\(x = 100\)
R’’(x) = −2 < 0, so the revenue is maximum at x = 100. The number of subscribers then is
\(500 - 100 = 400\)
Option (1), 100, is the increase in the charge (and the number of subscribers who leave), not the number who remain.