Match List-I with List-II

List-I	List-II
(A) $f\left(x\right)=-x$	(I) Not differentiable at x = -2 only
(B) $f\left(x\right)=-x+2$	(II) Not differentiable at x = 0 only
(C) $f\left(x\right)=|x^2-4|$	(III) Not differentiable at x = 2 only
(D) $f\left(x\right)=-x-2$	(IV) Not differentiable at x = 2, -2 only

Choose the correct answer from the options given below:

Correct Answer:
The correct option is (A) - (II), (B) - (I), (C) - (IV), (D) - (III).

Explanation:
To determine the points of non-differentiability, we analyze the absolute value functions represented in List-I. An absolute value function of the form $y=\left|g\right(x\left)\right|$ is typically continuous everywhere, but it is not differentiable at the points where the inner function $g\left(x\right)=0$ (resulting in sharp corners on the graph of the function).

Analysis of each function:

(A) $f\left(x\right)=\left|x\right|$:
The expression inside the absolute value is $x$.
Setting $x=0$, we find that the function has a sharp corner at $x=0$.
Thus, $f\left(x\right)$ is not differentiable at x = 0 only.
Hence, (A) matches with (II).

(B) $f\left(x\right)=|x+2|$:
The expression inside the absolute value is $x+2$.
Setting $x+2=0$ gives:
$x=-2$
Thus, the function is not differentiable at x = -2 only.
Hence, (B) matches with (I).

(C) $f\left(x\right)=|x^2-4|$:
The expression inside the absolute value is $x^2-4$.
Setting $x^2-4=0$ gives:
$(x-2)(x+2)=0\Rightarrow x=2\text{ or }x=-2$
Thus, the function has sharp corners and is not differentiable at x = 2, -2 only.
Hence, (C) matches with (IV).

(D) $f\left(x\right)=|x-2|$:
The expression inside the absolute value is $x-2$.
Setting $x-2=0$ gives:
$x=2$
Thus, the function is not differentiable at x = 2 only.
Hence, (D) matches with (III).

Combining all the matches:
(A) → (II), (B) → (I), (C) → (IV), (D) → (III)