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Question #41134

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \to \mathbb{R}$ be defined by $f(x) = \begin{cases} \frac{x^2}{\sin x} + x^2 \sin\left(\frac{1}{x}\right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}$ Let $g(x) = \frac{2}{\pi} - x$ . Consider the composite functions $H_1(x) = f(g(x))$ and $H_2(x) = g(f(x))$ . Which of the following statements is TRUE regarding the existence of local inverses for $H_1$ and $H_2$ in some neighborhood $(-\delta, \delta)$ of $x = 0$ (where $\delta > 0$ is sufficiently small)?

Options

A.
$H_1$ is invertible on $(-\delta, \delta)$ , but $H_2$ is not.
B.
$H_2$ is invertible on $(-\delta, \delta)$ , but $H_1$ is not.
C.
Both $H_1$ and $H_2$ are invertible on $(-\delta, \delta)$ .
D.
Neither $H_1$ nor $H_2$ is invertible on $(-\delta, \delta)$ .

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Composition Non-Commutativity|Inverse Existence Criterion|The L'Hôpital's Rule Misuse

Why This Question Is Tricky

Common Mistakes

This question tests the misconception that a positive derivative at a single point ( $f'(0) > 0$ ) guarantees local monotonicity and invertibility.

Students are baited into using L'Hôpital's rule to evaluate $f'(0)$ via the limit of the derivative, which leads to a non-existent limit, causing them to falsely conclude $f$ is non-differentiable.

If they correctly find $f'(0)=1$ , they often incorrectly assume $f(x)$ is invertible near $0$ .

Furthermore, the non-commutativity of composition is exploited: $H_2(x)$ evaluates $f$ near the pathological point $x=0$ , inheriting its non-monotonicity, whereas $H_1(x)$ evaluates $f$ near $x=2/\pi$ , where $f$ is smooth and strictly monotonic, making $H_1$ invertible.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Composition Non-Commutativity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free