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Question #41259

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined as $f(x) = \text{sgn}(x) \left[ \cos^{-1}\left( \frac{1-x^2}{1+x^2} \right) + \sin^{-1}\left( \frac{2x}{1+x^2} \right) \right]$ where $\text{sgn}(x)$ denotes the signum function. Let $S$ be the set of all points in $\mathbb{R}$ where $f(x)$ is not differentiable. Find the value of $\sum_{x \in S} (x^4 + 5x^2 + 2)$ .

Options

A.
16
B.
18
C.
20
D.
24

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

Variable Transformation Substitution|Chain Rule Complexity|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle variable transformations within inverse trigonometric functions while navigating the chain rule complexity introduced by a discontinuous outer multiplier, the signum function.

The 'Lethal Collision' occurs at $x=0$ .

Students often blindly apply standard differentiation formulas for ITFs, identifying $x=\pm 1$ as the only points of non-differentiability due to the domain splits of the inverse sine and cosine functions.

However, the presence of $\text{sgn}(x)$ creates a sharp corner at the origin because the left-hand and right-hand limits of the derivative evaluate to $0$ and $4$ respectively.

Failing to split the domain and manually check $x=0$ leads to missing this critical point, resulting in the trap answer 16.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Variable Transformation Subsitution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free