The demand function (in Rs.) for a product is given by \(P= 20 -0.25x\), where P is the price per unit and x is the number of units sold, then the price of one unit, when the revenue is maximized, is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Rs. 10
Revenue is price per unit times the number of units sold:
\(R = Px = (20 - 0.25x)x\)
\(R = 20x - 0.25x^2\)
For the maximum revenue, put \(\frac{dR}{dx} = 0\):
\( \displaystyle \frac{dR}{dx} = 20 - 0.5x = 0 \)
\(x = 40\)
\( \displaystyle \frac{d^2R}{dx^2} = -0.5 \lt 0 \), so this is a maximum.
The price per unit at x = 40 is
\(P = 20 - 0.25 \times 40 = 20 - 10 = 10\)
So the price is Rs. 10 per unit.
Why the other options are incorrect
Option (1) Rs. 20 — Incorrect. P = 20 only when x = 0, and then the revenue is 0.
Option (3) Rs. 30 — Incorrect. P = 30 would need x = −40, which is impossible.
Option (4) Rs. 40 — Incorrect. 40 is the number of units x at the maximum, not the price.