Let [x] denote the greatest integer func

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Continuity and Differentiability

Question:

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

List-I	List-II
(A) $\|x-1\|+\|x-2\|$	(I) is differentiable everywhere except at x = 0
(B) $ x-\|x\|$	(II) is continuous everywhere
(C) $x-[x]$	(III) is not differentiable at x = 1
(D) $x\|x\|$	(IV) is differentiable at x = 1

Choose the correct answer from the options given below:

Options:

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

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

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

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

Correct Answer:

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

Explanation:

The correct answer is Option (3) → (A) - (II), (B) - (I), (C) - (III), (D) - (IV)

A) |x−1| + |x−2| is a sum of modulus functions, hence continuous everywhere ⇒ (II)
(B) x − |x| is a piecewise function:
- 0 for x ≥ 0
- 2x for x < 0
  It has a sharp corner at x = 0, so it is differentiable everywhere except x = 0 ⇒ (I)
(C) x − [x] = {x} is the fractional part function, which is discontinuous at every integer, hence not differentiable at x = 1 ⇒ (III)
(D) x|x| behaves as:
- x² for x ≥ 0
- −x² for x < 0
  It is smooth at x = 1 and hence differentiable at x = 1 ⇒ (IV)