Question

Let [x] denote the greatest integer function. Then match List-I with List-II:

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

Hint:

When analyzing the behavior of piecewise functions like those involving modulus or greatest integer functions, it's crucial to check for continuity and differentiability at key points. Modulus functions are continuous, but piecewise functions like the greatest integer function or functions that involve absolute values may not be differentiable at certain points, often at the boundaries of the intervals. Understanding the behavior of these functions at critical points (such as \(x = 0\) or integer values) is key to determining their differentiability.

Answer 1

The Correct Option is C

For each function in List-I, analyze its behavior and match it with List-II:

For (A) \(|x - 1| + |x - 2|\): The modulus function \(|x - 1| + |x - 2|\) is continuous everywhere because modulus functions are inherently continuous. Match: (A) → (II).

For (B) \(x - |x|\): The function \(x - |x|\) is differentiable at \(x = 1\). This is because \(|x|\) is well-defined and continuous for all \(x\). Match: (B) → (I).

For (C) \(x - [x]\): The greatest integer function \([x]\) causes a lack of differentiability at all integers. Hence \(x - [x]\) is not differentiable at \(x = 1\). Match: (C) → (III).

For (D) \(x|x|\): The function \(x|x|\) is quadratic in behavior for both \(x>0\) and \(x<0\). Hence, it is differentiable everywhere except at \(x = 0\). Match: (D) → (IV).

\((A) – (II), (B) – (I), (C) – (III), (D) – (IV)\).

Answer 2

To match List-I with List-II correctly, we need to understand how the greatest integer function, denoted by [x], operates. The greatest integer function [x] represents the largest integer less than or equal to x. 
Analyzing each option in List-I and determining its corresponding expression from List-II:

• (A): Consider f(x) = [[x]]. For any real number x, [[x]] is effectively [x] itself, as the greatest integer function applied twice returns the same value. Thus, this corresponds to a general [x], which is expressed as (II): "0 if 0 ≤ x < 1 and −1 if −1 ≤ x < 0".
• (B): Consider f(x) = [2x]. This expression effectively multiplies x by 2 before applying the greatest integer function. Here, 0 ≤ x < 0.5 results in [2x] = 0, and 0.5 ≤ x < 1 results in [2x] = 1. This matches (I): "Integer function of 2x".
• (C): Consider f(x) = 2[x]. This expression multiplies the greatest integer directly by 2. For integers, this is simply even numbers corresponding to 2*[x]. Hence, this is (III): "2 times the greatest integer less than or equal to x".
• (D): Consider f(x) = [x + 0.5]. This shifts the function, and for 0 ≤ x < 0.5, [x + 0.5] = 0, and for 0.5 ≤ x < 1, [x + 0.5] = 1, thus covering all real numbers and matches (IV): "Periodically changing function".

Hence, the correct matching is: (A)-(II), (B)-(I), (C)-(III), (D)-(IV).