Which of the following statements are true?
A. Marginal cost represents the rate of change in total cost with respect to the number of items produced at an instant.
B. If C(x) represents the total cost function for producing x quantity of items, then marginal cost denoted by MC is given by \(MC=\displaystyle\frac{d}{dx}[C(x)]\)
C. The total cost function (in Rupees) is given by \(C(x) = 4x^2 + 100x - 90\). The marginal cost of producing 100 units is Rs. 900.
Choose the correct answer from the options given below
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A, B, C only
Marginal cost is the instantaneous rate of change of the total cost as the number of items produced changes, so it is the derivative of the cost function. For the given cost function:
\( C(x) = 4x^2 + 100x - 90 \)
\( MC = C'(x) = 8x + 100 \)
\( C'(100) = 8(100) + 100 = 900 \)
So MC is the rate of change of total cost with output, \(MC = \frac{dC}{dx}\), and here MC = Rs. 900 at x = 100.
Checking the statements
(A) Marginal cost represents the rate of change in total cost with respect to the number of items produced at an instant. — Correct. This is the definition of marginal cost.
(B) If C(x) represents the total cost function for producing x quantity of items, then \(MC=\frac{d}{dx}[C(x)]\) — Correct. The rate of change of C(x) with x is its derivative.
(C) For \(C(x) = 4x^2 + 100x - 90\), the marginal cost of producing 100 units is Rs. 900. — Correct. \(C'(100) = 8(100) + 100 = 900\).