Match List-I with List-II

List-I	List-II
Function f(x)	Points of Non-Differentiability
---	---
(A) $f(x) = \vert x\vert + 1$	(I) Not differentiable at $x = 3$ only
(B) $f(x) = \vert x - 3\vert$	(II) Not differentiable at $x = -3$ only
(C) $f(x) = \vert x + 3\vert$	(III) Not differentiable at $x = 3, -3$ only
(D) $f(x) = \vert x^2 - 9\vert$	(IV) Not differentiable at $x = 0$ only

Choose the correct answer from the options given below:

A function $f(x) = |g(x)|$ is not differentiable where $g(x) = 0$, because the absolute value creates a sharp corner at that point.

---

$f(x) = |x| + 1$

The expression inside the absolute value is $x$.

$x = 0$

Not differentiable at $x = 0$ only → (IV)

---

$f(x) = |x - 3|$

The expression inside the absolute value is $x - 3$.

$x - 3 = 0$

$x = 3$

Not differentiable at $x = 3$ only → (I)

---

$f(x) = |x + 3|$

The expression inside the absolute value is $x + 3$.

$x + 3 = 0$

$x = -3$

Not differentiable at $x = -3$ only → (II)

---

$f(x) = |x^2 - 9|$

The expression inside the absolute value is $x^2 - 9$.

$x^2 - 9 = (x - 3)(x + 3) = 0$

$x = 3$ or $x = -3$

Not differentiable at $x = 3, -3$ only → (III)

---

(A) → (IV)

(B) → (I)

(C) → (II)

(D) → (III)

Therefore, the correct answer is (A)-(IV), (B)-(I), (C)-(II), (D)-(III).