Match List-I with List-II

Choose the correct answer from the options given below:

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

The correct answer is Option (2) → (A)-(II), (B)-(I), (C)-(IV), (D)-(III)

$(A)\ f(x)=|x|$

$|x|=\begin{cases}x,&x\ge 0\\-x,&x<0\end{cases}$

Left derivative at $x=0=-1$, right derivative at $x=0=1$

$f'(0^-)\ne f'(0^+)$

Not differentiable at $x=0$ only

$(A)\rightarrow(II)$

$(B)\ f(x)=|x+2|$

Corner point when $x+2=0$

$x=-2$

Not differentiable at $x=-2$ only

$(B)\rightarrow(I)$

$(C)\ f(x)=|x^2-4|$

$x^2-4=0 \Rightarrow x=\pm2$

Sign of $x^2-4$ changes at $x=2,-2$

Derivative is discontinuous at both points

Not differentiable at $x=2,-2$

$(C)\rightarrow(IV)$

$(D)\ f(x)=|x-2|$

Corner point when $x-2=0$

$x=2$

Not differentiable at $x=2$ only

$(D)\rightarrow(III)$

Final Matching: (A)-(II), (B)-(I), (C)-(IV), (D)-(III).