Consider the function $f:(0, \infty) \rightarrow(-\infty, \infty)$ given by \[ f(\mathrm{x})=\sqrt{\mathrm{x}} \log _{\mathrm{e}}(\mathrm{x})-\mathrm{x}+1 \] Then which one of the following statements is TRUE ?

(D) The function $f$ has NEITHER a point of local maximum NOR a point of local minimum in the interval $(0, \infty)$

(A) The derivative of the function $f$ is decreasing in the interval $(0,1)$

(B) The function $f$ has a local maximum at some point $\mathrm{a} \in(0, \infty)$

(C) The function $f$ has a local minimum at some point $\mathrm{b} \in(0, \infty)$

JEE Advanced · 2026 · 17 May 2026 · Paper 1 · Application of Derivatives

Consider the function

f : (0, \infty) \to (- \infty, \infty)

given by

f (x) = x lo g_{e} (x) - x + 1

Then which one of the following statements is TRUE ?

(A)The derivative of the function $f$ is decreasing in the interval $(0, 1)$
(B)The function $f$ has a local maximum at some point $a \in (0, \infty)$
(C)The function $f$ has a local minimum at some point $b \in (0, \infty)$
(D)The function $f$ has NEITHER a point of local maximum NOR a point of local minimum in the interval $(0, \infty)$ ✓ Correct

Answer: (D)

Given

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

.

First derivative:

f^{'} (x) = \frac{1}{x} + \frac{ln x}{2 x} - 1 = \frac{ln x + 2 - 2 x}{2 x}

Second derivative:

f^{''} (x) = \frac{- ln x}{4 x ^{3/2}}

For

f^{'}

to be decreasing, we need

f^{''} (x) \leq 0

, which gives

ln x \geq 0

, i.e.,

x \geq 1

. So

f^{'}

is decreasing on

(1, \infty)

, not

(0, 1)

. Option (A) is false.

Let

g (x) = ln x + 2 - 2 x

. Then

g (1) = 0

and

g^{'} (x) = \frac{1}{x} - \frac{1}{x} = \frac{1 - x}{x}

.

For

x \in (0, 1)

g^{'} (x) > 0

, so

g

is increasing. Since

x < 1

, we have

g (x) < g (1) = 0

.

For

x \in (1, \infty)

g^{'} (x) < 0

, so

g

is decreasing. Since

x > 1

, we have

g (x) < g (1) = 0

.

Therefore

g (x) < 0

for all

x \in (0, \infty) ∖ {1}

, which means

f^{'} (x) < 0

for all

x \neq = 1

. Thus

f

is strictly decreasing on

(0, \infty)

and has no local extrema.

Option (D) is correct.

JEE Advanced · 2026 · 17 May 2026 · Paper 1 · Single correct option