Consider the following three statements for the function $f : (0, \infty) \to ℝ$ defined by
$f (x) =$ $| \log_{e} x |$ $-$ $| x - 1 |$ $:$

Question

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

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

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

Then. [2026]

Smart Achievers · Accepted Answer

(1)

$f (x) =$ $| l n x |$ $-$ $| x - 1 |$

$=$ ${\begin{matrix} l n x - (x - 1) & x \geq 1 \\ - l n x + (x - 1) & 0 < x < 1 \end{matrix}$

$=$ ${\begin{matrix} l n x - x + 1 & x \geq 1 \\ - l n x + x - 1 & 0 < x < 1 \end{matrix}$

$f' (x)$ $=$ $(\begin{matrix} \frac{1}{x} - 1 & x \geq 1 \\ - \frac{1}{x} + 1 & 0 < x < 1 \end{matrix}$

$f' (1^{+}) = f' (1^{-}) = 0 \Rightarrow f (x) is differentiable \forall x > 0$

$f' (x) < 0 \forall x > 1$

$f' (x) < 0 \forall 0 < x < 1$

$\Rightarrow f (x) is decreasing \forall x \in (0, \infty)$

$Option (1)$

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