Consider the following three statements for the function f:(0, ∞) arrow ℝ defined by f(x)=| _ e x|-|x-1| : (I) f is differentiable at all x>0 . (II) f is increasing in (0,1) . (III) f is decreasing in (1, ∞) . Then.

JEE Main 202624 JanuaryEvening Shift

Exam: JEE Main
Year: 2026
Date: 24 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Application of Derivatives

JEE Main 2026 · 24 January · Evening Shift · Mathematics · Application of Derivatives

Consider the following three statements for the function f:(0, ∞)...

Consider the following three statements for the function

f:(0, \infty) \rightarrow \mathbb{R}

defined by

f(x)=\left|\log _{\mathrm{e}} x\right|-|x-1|

:
(I)

f

is differentiable at all

x>0

.
(II)

f

is increasing in

(0,1)

.
(III)

f

is decreasing in

(1, \infty)

.
Then.

Options

AAll (I), (II) and (III) are TRUE.
BOnly (II) and (III) are TRUE.
COnly (I) is TRUE.
DOnly (I) and (III) are TRUE.

Correct answer

D. Only (I) and (III) are TRUE.

Step-by-step solution

For

0 < x < 1

f(x) = -\ln x + x - 1

f'(x) = -\frac{1}{x} + 1 = \frac{x-1}{x} < 0

(decreasing).
For

x > 1

f(x) = \ln x - x + 1

f'(x) = \frac{1}{x} - 1 = \frac{1-x}{x} < 0

(decreasing).
At

x = 1

: LHD

= \frac{1-1}{1} = 0

, RHD

= \frac{1-1}{1} = 0

. So

f

is differentiable at

x = 1

.
(I) TRUE:

f

is differentiable for all

x > 0

.
(II) FALSE:

f

is decreasing in

(0, 1)

.
(III) TRUE:

f

is decreasing in

(1, \infty)

.
Only (I) and (III) are TRUE.

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