The total revenue (in Rs.) from the sale of x units of a product is \(R(x) = 48x - x^3\), \(x \gt 0\). Consider the following statements:
(A) The marginal revenue when 2 units are sold is Rs. 36.
(B) The total revenue is increasing for 0 < x < 4.
(C) The maximum total revenue is Rs. 128.
(D) The average revenue when 2 units are sold is Rs. 40.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (C) only
The marginal revenue and the average revenue are
\(MR = R'(x) = 48 - 3x^2\)
\( \displaystyle AR = \frac{R(x)}{x} = 48 - x^2 \)
MR = 0 when \(x^2 = 16\), that is x = 4. MR is positive for 0 < x < 4 and negative for x > 4, so R is greatest at x = 4:
\(R(4) = 192 - 64 = 128\)
So MR(2) = 36, R increases for 0 < x < 4, the maximum revenue is Rs. 128, and AR(2) = 44.
Checking the statements
(A) The marginal revenue when 2 units are sold is Rs. 36. — Correct. MR(2) = 48 − 12 = 36.
(B) The total revenue is increasing for 0 < x < 4. — Correct. \(R'(x) = 3(16 - x^2) \gt 0\) for 0 < x < 4.
(C) The maximum total revenue is Rs. 128. — Correct. The maximum is at x = 4: R(4) = 192 − 64 = 128.
(D) The average revenue when 2 units are sold is Rs. 40. — Incorrect. AR(2) = 48 − 4 = 44.