Mathematics Question from JEE Main 2026 · 24 January, Shift 2

Consider the following three statements for the function $f : (0,\infty) \rightarrow \mathbb{R}$ defined by

$$f(x) = \left| \log_e x \right| - |x - 1|$$

:

(I) $f$ is differentiable at all $x > 0$.

(II) $f$ is increasing in $(0,1)$.

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

Then,