Consider the following three statements for the function f:(0, ∞) → R defined by f(x)=| log_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.

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

Consider the following three statements for the function f:(0, ∞) → R defined by f(x)=|…

Mathematics · 2026

JEE Main · 24 January 2026, Shift 2 · Q17

Consider the following three statements for the function $\displaystyle f:(0, \infty) \rightarrow \mathbb{R}$ defined by $\displaystyle f(x)=\left|\log _{\mathrm{e}} x\right|-|x-1|$ :

(I)

$\displaystyle f$ is differentiable at all $\displaystyle x>0$.

(II)

$\displaystyle f$ is increasing in $\displaystyle ( 0,1 )$.

(III)

$\displaystyle f$ is decreasing in $\displaystyle (1, \infty)$.

Then.

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Consider the following three statements for the function f:(0, ∞) → R defined by f(x)=| log_e x|-|x-1|: (I) f is differentiable at all x>0. (II) f is…

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