Match List – I with List – II.

Choose the correct answer from the options given below:

A - IV, B - I, C - II, D - III

The correct answer is Option (3) → A - IV, B - I, C - II, D - III

$\text{(A)}\; y=x^3\log x$

$y'=3x^2\log x + x^2$

$y''=6x\log x + 3x + 2x = x(6\log x + 5) \Rightarrow \text{(IV)}$

$\text{(B)}\; y=e^{5x}$

$y''=25e^{5x} \Rightarrow \text{(I)}$

$\text{(C)}\; y=x^x$

$y'=x^x(1+\log x) \Rightarrow \text{(II)}$

$\text{(D)}\; y=at^2,\; x=2at$

$\frac{d^2y}{dx^2}=\frac{1}{2a t^3}(-1) = -\frac{1}{2at^3} \Rightarrow \text{(III)}$

A–IV,\; B–I,\; C–II,\; D–III