A cable network provider in a small town has 500 subscriber and he used to collect Rs. 300 per month from each subscriber. He proposes o increase the monthly charges and it is believed from the past experience that for every increase of Rs. 1, one subscriber will discontinue the service.
Based on the above information answer the following question:
What is the increase in the changes per subscriber that yields maximum revenue?
Based on the above information answer the following question:
What is the increase in the changes per subscriber that yields maximum revenue?
- 100
- 200
- 300
- 400
The Correct Option is A
Solution and Explanation
Let company increases the annual subscription by \(Rs\ x\).
So, x subscribers will discontinue the service.
Total revenue of company after the increment
\(R(x) =(500−x)(300+x)\)
\(R(x) =1500000+500x−300x−x^2\)
\(R(x) = −x^2+200x+150000\)
Differentiate both sides w.r.t, x
\(R^′(x)=−2x+200\)
Now, \(R^′(x)=0\)
\(2x=200\)
\(x=100\)
∴ \(R^{''}(x)=−2<0\)
R(x) is maximum when \(x = 100\).
Therefore, the company should increase the subscription fee by \(Rs.\ 100\), so that it has maximum revenue.
So, the correct option is (A): \(100\)
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Concepts Used:
Application of Derivatives
Various Applications of Derivatives-
Rate of Change of Quantities:
If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by
\(\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)
This is also known to be as the Average Rate of Change.
Increasing and Decreasing Function:
Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).
- If for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≤ f(x2); then the function f(x) is known as increasing in this interval.
- Likewise, if for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≥ f(x2); then the function f(x) is known as decreasing in this interval.
- The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.
Read More: Application of Derivatives