Consider the following three statements for the function $f:\left(0,∞\right)\to ℝ$ defined by $f\left(x\right)=\text{ }∣{\text{log}}_{e}x∣−|x−1∣:$

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

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

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

[JEE Main 2026, 24 Jan (Shift 2)]

Question

Consider the following three statements for the function $f:\left(0,∞\right)\to ℝ$ defined by $f\left(x\right)=\text{ }∣{\text{log}}_{e}x∣−|x−1∣:$

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

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

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

[JEE Main 2026, 24 Jan (Shift 2)]

YESPYQ · Accepted Answer

Correct answer: b) Only (I) and (III) are TRUE.. $f(x)=|lnx|−|x−1|$ $=\left{\begin{matrix}lnx−(x−1) & x\geq 1 \ −lnx+(x−1) & 0 \(=\left{\begin{matrix}lnx−x+1 & x\geq 1 \ −lnx+x−1 & 0 \({f}^{'}\left(x\right)=\left{\begin{matrix}\frac{1}{x}-1 & x\geq 1 \ -\frac{1}{x}+1 & 0 \({f}^{'}\left({1}^{+}\right)={f}^{'}\left({1}^{−}\right)=0$ $\Rightarrow f(x)$ is differentiable $\forall x>0$. $f'(x) 1\ f'(x) \(\Rightarrow f(x)$ is decreasing $\forall x\in (0,\infty )$.

Consider the following three statements for the function \(f:\left(0,∞\right)\to ℝ\) defined by \(f\left(x\right)=\text{…

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