Question

Match List-I with List-II

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

Choose the correct answer from the options given below:

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

Hint:

To quickly find points of non-differentiability for an absolute value function f(x) = |g(x)|, solve the equation g(x) = 0. The roots of this equation are the candidates for points where the function is not differentiable, especially if g'(x) is not zero at those roots.

Answer 1

The Correct Option is B

Step 1: Conceptual Understanding:
A function of the form f(x) = |g(x)| is generally not differentiable at points where g(x) = 0, due to potential sharp corners or cusps in the graph. Identify these points for each function in List-I.

Step 3: Detailed Analysis:
Analyze each function from List-I.

(A) f(x) = |x|
Non-differentiability occurs when the expression inside the absolute value is zero: \(x = 0\).
Thus, f(x) = |x| is not differentiable solely at x = 0. This corresponds to (II).

(B) f(x) = |x + 2|
Set the inner expression to zero: \(x + 2 = 0 \implies x = -2\).
Therefore, f(x) = |x + 2| is not differentiable solely at x = -2. This matches with (I).

(C) f(x) = $|x^2 - 4|$
Set the inner expression to zero: \(x^2 - 4 = 0 \implies (x - 2)(x + 2) = 0\).
The solutions are x = 2 and x = -2.
Consequently, f(x) = $|x^2 - 4|$ is not differentiable at x = 2 and x = -2. This matches with (IV).

(D) f(x) = |x - 2|
Set the inner expression to zero: \(x - 2 = 0 \implies x = 2\).
Hence, f(x) = |x - 2| is not differentiable solely at x = 2. This matches with (III).

Step 4: Conclusion:
The correct pairings are:
(A) $\rightarrow$ (II)
(B) $\rightarrow$ (I)
(C) $\rightarrow$ (IV)
(D) $\rightarrow$ (III)
This corresponds to option (2).