The marginal revenue function of a firm is \(MR = 20 - 4x\), where x is the number of units sold, and the revenue is zero when no unit is sold. The total revenue function R(x) and the demand function p are
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(R(x) = 20x - 2x^2\) and \(p = 20 - 2x\)
The marginal revenue is the derivative of the total revenue, so the total revenue is the integral of MR:
\( \displaystyle R(x) = \int (20 - 4x)\,dx = 20x - 2x^2 + k \)
Since R(0) = 0, the constant k = 0:
\(R(x) = 20x - 2x^2\)
The revenue is price times quantity, \(R = px\), so the demand function is
\( \displaystyle p = \frac{R(x)}{x} = 20 - 2x \)
Why the other options are incorrect
Option (2) \(R(x) = 20x - 4x^2\) and \(p = 20 - 4x\) — Incorrect. The derivative of \(20x - 4x^2\) is \(20 - 8x\), not \(20 - 4x\); the power rule needs the division by 2.
Option (3) \(R(x) = 20x - 2x^2\) and \(p = 20 - 4x\) — Incorrect. The revenue is right, but the price is \(\frac{R}{x} = 20 - 2x\); the marginal revenue has been taken as the price.
Option (4) \(R(x) = 10x - x^2\) and \(p = 10 - x\) — Incorrect. The derivative of \(10x - x^2\) is \(10 - 2x\), which is only half of the given MR.