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Question #40293

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: D \to \mathbb{R}$ be a function defined by $f(x) = \frac{\ln(1 + x^2)}{\text{sgn}(x) + [x]}$ where $D \subset (-1, 1)$ is the maximal domain of $f$ in the interval $(-1, 1)$ . Here, $[x]$ denotes the greatest integer less than or equal to $x$ , and $\text{sgn}(x)$ denotes the signum function. Let $S$ be the set of all critical points of $f(x)$ in $D$ . Which of the following represents the set $S$ ?

Options

A.
{0}
B.
\emptyset
C.
{-1/2, 1/2}
D.
(-1, 0) \cup (0, 1)

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

The Empty Set Trap|Critical Point vs Stationary Point|Signum Function Discontinuity|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

The question targets the 'Empty Set Trap' by exploiting the formal definition of a critical point.

Students routinely calculate the derivative, find $x=0$ as a root of the numerator, and observe that the function is discontinuous (or non-differentiable) at $x=0$ .

Consequently, they classify $x=0$ as a critical point, selecting the singleton set {0}.

However, the mathematical definition strictly requires a critical point to be in the domain of the function.

Because the denominator evaluates to zero at $x=0$ , it creates a domain hole.

Thus, $x=0$ is excluded, leaving the set of critical points entirely empty.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	The Empty Set Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free