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

Analyze the behavior of each function in List-I and match it with the corresponding option in List-II:

For (A) \(|x - 1| + |x - 2|\): This is a sum of continuous functions, making it continuous everywhere. Match: (A) → (II).

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

For (C) \(x - [x]\): The greatest integer function \([x]\) introduces points of non-differentiability at all integer values. Therefore, \(x - [x]\) is not differentiable at \(x = 1\). Match: (C) → (III).

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

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