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Question #40951

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all real values of $x$ for which the function $f(x) = \left| x^2 - 4 \right| e^{-|x|}$ is strictly increasing. Which of the following represents the set $S$ ?

Options

A.
$(-\infty, -1-\sqrt{5}) \cup (-2, 0) \cup (2, 1+\sqrt{5})$
B.
$(-1-\sqrt{5}, -2) \cup (0, 2) \cup (1+\sqrt{5}, \infty)$
C.
$(1-\sqrt{5}, 0) \cup (2, 1+\sqrt{5})$
D.
$(-\infty, -2) \cup (0, 2)$

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

Chain Rule Complexity|Increasing/Decreasing Strictness|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the chain rule and product rule on functions containing multiple modulus operators.

The primary trap lies in the sign dependency of the modulus arguments.

Students often attempt to differentiate $|g(x)|$ globally using the signum function without carefully partitioning the domain at the critical points ( $x = \pm 2$ and $x = 0$ ).

This rote differentiation leads to incorrect signs in the derivative across different intervals.

Recognizing the non-differentiability at these roots is crucial, as it invalidates a global monotonicity analysis and forces the necessity of piecewise differentiation, a classic discriminator in JEE Advanced.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Chain Rule Complexity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free