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Question #63611

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all points of local extrema of the function $f(x) = \sin^{-1}\left( \frac{1 - x^2}{1 + x^2} \right) + \frac{2x}{5}$ on $\mathbb{R}$ . If $n(S)$ denotes the number of elements in $S$ , then the value of $3^{n(S)} + \sum_{x \in S} x$ is equal to:

Options

A.
11
B.
9
C.
5
D.
7

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Algebraic-Trigonometric Graphical Mapping|Tangent Half-Angle Substitution|First Derivative Test Trap

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the critical evaluation of Inverse Trigonometric Functions (ITFs) and the First Derivative Test at non-differentiable boundaries.

Students commonly fall into the trap of blindly applying the identity $\sin^{-1}\left(\frac{1-x^2}{1+x^2}\right) = \frac{\pi}{2} - 2\tan^{-1}x$ for all $x \in \mathbb{R}$ , ignoring the absolute value requirement $|2\tan^{-1}x|$ .

This false assumption yields a smooth rational derivative $\frac{2x^2-8}{5(1+x^2)}$ , producing a deceptive local maximum at $x = -2$ .

By correctly identifying the piecewise nature of the function, the false stationary point is eliminated, and the true local maximum is revealed at the non-differentiable corner point $x = 0$ , validating the necessity of checking boundary conditions in substitution domains.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Algebraic-Trigonometric Graphical Mapping
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free