Match List-I with List-II

List-I	List-II
(A) Marginal average cost if cost function $C(x) = \frac{50}{\sqrt{x}}$	(I) $50\sqrt{x}$
(B) Marginal average cost if cost function $C(x) = 50\sqrt{x}$	(II) $-\frac{75}{x^2\sqrt{x}}$
(C) Revenue function if demand function $P=\frac{50}{\sqrt{x}}$	(III) $\frac{-25}{x\sqrt{x}}$
(D) Marginal revenue if demand function $P=50\sqrt{x}$	(IV) $75\sqrt{x}$

Choose the correct answer from the options given below:

Average Cost $AC(x) = \frac{C(x)}{x}$

Marginal Average Cost $= \frac{d(AC)}{dx}$

Revenue $R(x) = P \times x$

Marginal Revenue $MR = \frac{dR}{dx}$

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For $C(x) = \frac{50}{\sqrt{x}}$:

Average Cost:

$AC(x) = \frac{C(x)}{x}$

$= \frac{50/\sqrt{x}}{x}$

$= \frac{50}{x\sqrt{x}}$

$= 50x^{-3/2}$

Marginal Average Cost:

$\frac{d(AC)}{dx} = \frac{d}{dx}(50x^{-3/2})$

$= 50 \times \left(-\frac{3}{2}\right) \times x^{-5/2}$

$= -75x^{-5/2}$

$= -\frac{75}{x^2\sqrt{x}}$

(A) matches with (II)

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For $C(x) = 50\sqrt{x}$:

Average Cost:

$AC(x) = \frac{50\sqrt{x}}{x}$

$= \frac{50}{\sqrt{x}}$

$= 50x^{-1/2}$

Marginal Average Cost:

$\frac{d(AC)}{dx} = 50 \times \left(-\frac{1}{2}\right) \times x^{-3/2}$

$= -25x^{-3/2}$

$= -\frac{25}{x\sqrt{x}}$

(B) matches with (III)

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For demand function $P = \frac{50}{\sqrt{x}}$:

Revenue function:

$R(x) = P \times x$

$= \frac{50}{\sqrt{x}} \times x$

$= \frac{50x}{\sqrt{x}}$

$= 50\sqrt{x}$

(C) matches with (I)

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For demand function $P = 50\sqrt{x}$:

Revenue function:

$R(x) = P \times x$

$= 50\sqrt{x} \times x$

$= 50x^{3/2}$

Marginal Revenue:

$MR = \frac{dR}{dx}$

$= 50 \times \frac{3}{2} \times x^{1/2}$

$= 75\sqrt{x}$

(D) matches with (IV)

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Final Matching:

(A) → (II): $-\frac{75}{x^2\sqrt{x}}$

(B) → (III): $-\frac{25}{x\sqrt{x}}$

(C) → (I): $50\sqrt{x}$

(D) → (IV): $75\sqrt{x}$