If a revenue function is given by \(R(x) = 2027x - 1013x^2 - 675x^3\), then the marginal revenue function (MR) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(MR = 2027 - 2026x - 2025x^2\)
The marginal revenue is the derivative of the revenue function:
\( \displaystyle MR = \frac{dR}{dx} = \frac{d}{dx}\left(2027x - 1013x^2 - 675x^3\right) \)
\(= 2027 - 2 \times 1013x - 3 \times 675x^2\)
\(= 2027 - 2026x - 2025x^2\)
Why the other options are incorrect
Option (1) \(MR = 2027 - 1013x - 675x^2\) — Incorrect. This is \(\frac{R(x)}{x}\), the average revenue, not the marginal revenue.
Option (2) \(MR = 2026 - 4050x\) — Incorrect. The derivative of the cubic \(R(x)\) is a quadratic in x, with an \(x^2\) term.
Option (4) \(MR = 1013x - 2025x^2\) — Incorrect. The term 2027x of R(x) gives the constant 2027 in MR, which is missing here.