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Question #41159

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) + \frac{x^3}{3} - x$ . Find the total number of points of local extrema of $f(x)$ .

Options

A.
2
B.
3
C.
4
D.
6

Check Your Answer

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

Principal Value Branch Boundaries|Increasing/Decreasing Strictness|Second Derivative Test Limitations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to identify local extrema at points of non-differentiability, specifically at the boundaries of the principal value branches of inverse trigonometric functions.

A common trap is relying solely on the condition $f'(x) = 0$ and the second derivative test.

Students falling for this trap will only find the extrema at $x = \pm 3^{1/4}$ and completely miss the 'lethal collision' at $x = \pm 1$ , where the derivative is undefined but the function remains continuous and changes monotonicity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Principal Value Branch Boundaries
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free