Applications of Derivatives
Differentiability and Monotonicity — Absolute Value Function
nta_pyq_2026_jan
Grade 12
Question:
Consider the following three statements for the function $f:(0,\infty)\to\mathbb{R}$ 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 $(1,\infty)$.
Then,
All (I), (II) and (III) are TRUE.
Only (II) and (III) are TRUE.
Only (I) is TRUE.
Only (I) and (III) are TRUE.
Step-by-Step Solution
Key Concept: For $0<x<1$: $f(x)=-\ln x-(1-x)$, $f'(x)=-1/x+1=(x-1)/x<0$ (decreasing). For $x>1$: $f(x)=\ln x-(x-1)$, $f'(x)=1/x-1=(1-x)/x<0$ (decreasing). At $x=1$: LHD $=$ RHD $=0$ so $f$ is differentiable.
$f$ differentiable everywhere, decreasing on both $(0,1)$ and $(1,\infty)$. Only (I) and (III) true.
Correct Answer: 4