Match List-I with List-II
| List-I | List-II | ||
| (A) | Marginal average cost if cost function \(C(x) = \displaystyle\frac{50}{\sqrt{x}}\) | (I) | \(50\sqrt{x}\) |
| (B) | Marginal average cost if cost function \(C(x) = 50\sqrt{x}\) | (II) | \(-\displaystyle\frac{75}{x^2\sqrt{x}}\) |
| (C) | Revenue function if demand function \(P = \displaystyle\frac{50}{\sqrt{x}}\) | (III) | \(\displaystyle\frac{-25}{x\sqrt{x}}\) |
| (D) | Marginal revenue if demand function \(P = 50\sqrt{x}\) | (IV) | \(75\sqrt{x}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(II), (B)-(III), (C)-(I), (D)-(IV)
The average cost is \(\frac{C(x)}{x}\), and the marginal average cost is its derivative. The revenue is \(R = Px\), and the marginal revenue is \(\frac{dR}{dx}\).
(A) Marginal average cost if \(C(x) = \frac{50}{\sqrt{x}}\)
This matches (II) \(-\frac{75}{x^2\sqrt{x}}\):
\( \displaystyle \frac{C(x)}{x} = \frac{50}{x\sqrt{x}} = 50x^{-3/2} \)
\( \displaystyle \frac{d}{dx}\left(50x^{-3/2}\right) = -75x^{-5/2} = -\frac{75}{x^2\sqrt{x}} \)
(B) Marginal average cost if \(C(x) = 50\sqrt{x}\)
This matches (III) \(\frac{-25}{x\sqrt{x}}\):
\( \displaystyle \frac{C(x)}{x} = \frac{50\sqrt{x}}{x} = 50x^{-1/2} \)
\( \displaystyle \frac{d}{dx}\left(50x^{-1/2}\right) = -25x^{-3/2} = \frac{-25}{x\sqrt{x}} \)
(C) Revenue function if \(P = \frac{50}{\sqrt{x}}\)
This matches (I) \(50\sqrt{x}\):
\( \displaystyle R = Px = \frac{50}{\sqrt{x}} \times x = 50\sqrt{x} \)
(D) Marginal revenue if \(P = 50\sqrt{x}\)
This matches (IV) \(75\sqrt{x}\):
\(R = Px = 50x^{3/2}\)
\( \displaystyle \frac{dR}{dx} = 50 \times \frac{3}{2}x^{1/2} = 75\sqrt{x} \)
So the matching is (A)-(II), (B)-(III), (C)-(I), (D)-(IV).