If the marginal revenue function of a commodity is given by \( 6-4x^3+3x^2 \), then the demand function is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 6-x^3+x^2 \)
The revenue function R is the integral of the marginal revenue, with R = 0 when x = 0 (no sales, no revenue):
\( \displaystyle R=\int\left(6-4x^3+3x^2\right)dx \)
\( \displaystyle =6x-x^4+x^3+C \)
\( \displaystyle R(0)=0\ \Rightarrow\ C=0 \)
\( \displaystyle R=6x-x^4+x^3 \)
Revenue = price × quantity, R = px, so the demand (price) function is R divided by x:
\( \displaystyle p=\frac{R}{x}=\frac{6x-x^4+x^3}{x} \)
\( \displaystyle =6-x^3+x^2 \)
Why the other options are incorrect
Option (1) \( 6x-x^4+x^3 \) — Incorrect. This is the revenue function R; the demand function is R divided by x.
Option (3) \( 6-4x^3 \) — Incorrect. This is the marginal revenue without its \( 3x^2 \) term, not R divided by x.
Option (4) \( 3x^2-4x^3 \) — Incorrect. This is the marginal revenue without its constant term, not the demand function.