Question

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,

• A. All (I), (II) and (III) are TRUE.
• B. Only (II) and (III) are TRUE.
• C. Only (I) and (III) are TRUE.
• D. Only (I) is TRUE.

Hint:

When absolute values are involved, always split the domain and analyze each interval separately.

Answer 1

The Correct Option is C

Step 1: Differentiability of $f(x)$.
The function is composed of absolute value expressions $|\log x|$ and $|x-1|$. Both are differentiable for all $x>0$ except possibly at $x=1$. At $x=1$, the left-hand and right-hand derivatives of both terms exist and are finite. Hence, $f(x)$ is differentiable for all $x>0$. Therefore, statement (I) is true.
Step 2: Monotonicity in $(0,1)$.
For $0<x<1$, we have $|\log x|=-\log x$ and $|x-1|=1-x$. Thus, \[ f(x) = -\log x - (1-x). \] Differentiating, \[ f'(x) = -\frac{1}{x} + 1<0 \quad \text{for } 0<x<1. \] Hence, $f(x)$ is decreasing in $(0,1)$, not increasing. Statement (II) is false.
Step 3: Monotonicity in $(1,\infty)$.
For $x>1$, we have $|\log x|=\log x$ and $|x-1|=x-1$. Thus, \[ f(x) = \log x - (x-1). \] Differentiating, \[ f'(x) = \frac{1}{x} - 1<0 \quad \text{for } x>1. \] Hence, $f(x)$ is decreasing in $(1,\infty)$. Statement (III) is true.
Step 4: Conclusion.
Only statements (I) and (III) are true.