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:
Answer & explanation
Correct answer: option 3
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)