Question

A private telephone company serving a small community makes a profit of ₹ 12 per subscriber, if it has 725 subscribers. It decides to reduce the rate by a fixed sum for each subscriber over 725, thereby reducing the profit by 1 paise per subscriber. Thus, there will be profit of ₹ 11.99 on each of the 726 subscribers, ₹ 11.98 on each of the 727 subscribers etc. The number of subscribers which will give the company, the maximum profit, is:

• A. $961$
• B. $962$
• C. $963$
• D. None of these

Hint:

When profit per unit decreases linearly with quantity, total profit is quadratic $P(q)=aq-bq^2$. The vertex is at $q=\dfrac{a}{2b}$. For integer $q$, check the two integers around the vertex; symmetry often makes them tie.

Answer 1

The Correct Option is B

Step 1 (Model the profit per subscriber).
Let $n$ be the number of subscribers. For every extra subscriber above $725$, the profit per subscriber drops by ₹ $0.01$ \(\textit{for all subscribers}\). Hence \[ p(n)=12-0.01\,(n-725)=12-0.01n+7.25=\boxed{\,19.25-0.01n\,}\;(\text{rupees per subscriber}). \] 

Step 2 (Total profit as a function of $n$).
\[ P(n)=n\,p(n)=n(19.25-0.01n)=\boxed{-0.01n^2+19.25\,n}, \] which is a concave quadratic (opens downward), so it has a unique maximum at its vertex. 

Step 3 (Continuous maximizer / vertex).
For $P(n)=-an^2+bn$, the vertex is at $n^\star=\dfrac{b}{2a}$. Here $a=0.01,\ b=19.25$: \[ n^\star=\frac{19.25}{2\cdot 0.01}=\frac{19.25}{0.02}=962.5. \] Since $n$ must be an integer, only the nearest integers $962$ and $963$ can maximize $P$. 

Step 4 (Show the tie rigorously).
A discrete check via first difference: \[ P(n{+}1)-P(n)=\left[-0.01(n{+}1)^2+19.25(n{+}1)\right]-\left[-0.01n^2+19.25n\right] =19.24-0.02n. \] At $n=962$, $P(963)-P(962)=19.24-0.02\cdot 962=0 \Rightarrow P(963)=P(962)$ (tie).
Also $P(962)-P(961)=19.24-0.02\cdot 961 > 0$, so $P(962) > P(961)$. 

Step 5 (Numerical confirmation).
\[ \begin{aligned} P(962)&=962\big(19.25-9.62\big)=962\times 9.63=₹ 9264.06, \\ P(963)&=963\big(19.25-9.63\big)=963\times 9.62=₹ 9264.06. \end{aligned} \] Both equal and exceed the profit at neighboring integers. \[ \boxed{\text{Maximum total profit occurs at } n=962 \text{ \textit{and} } n=963.} \]