Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(IV), (B)-(III), (C)-(II), (D)-(I)

The correct answer is Option (4) → (A)-(IV), (B)-(III), (C)-(II), (D)-(I)

Given curves and slopes to find at x = 4:

(A) $y = \sqrt{x^3} = x^{3/2}$

$\frac{dy}{dx} = \frac{3}{2} x^{1/2}$

At $x = 4$: $\frac{dy}{dx} = \frac{3}{2} \cdot 2 = 3$ → (A)-(IV)

(B) $y = \sqrt{x} = x^{1/2}$

$\frac{dy}{dx} = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}$

At $x = 4$: $\frac{dy}{dx} = \frac{1}{2*2} = 1/4$ → (B)-(III)

(C) $y = x^3 - 47x$

$\frac{dy}{dx} = 3x^2 - 47$

At $x = 4$: $\frac{dy}{dx} = 3*16 - 47 = 48 - 47 = 1$ → (C)-(II)

(D) $xy = 16 \Rightarrow y = 16/x$

$\frac{dy}{dx} = -16/x^2$

At $x = 4$: $\frac{dy}{dx} = -16/16 = -1$ → (D)-(I)