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Question #41362

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = |x^2 - 4|^{\ln |x^2 - 4|}$ for $x \in \mathbb{R} \setminus \{-2, 2\}$ . The set of all values of $x$ for which $f(x)$ is strictly increasing is:

Options

A.
$(-\sqrt{5}, -2) \cup (-\sqrt{3}, 0) \cup (\sqrt{3}, 2) \cup (\sqrt{5}, \infty)$
B.
$(-\sqrt{5}, -2) \cup (-2, -\sqrt{3}) \cup (0, \sqrt{3}) \cup (\sqrt{5}, \infty)$
C.
$(-\infty, -\sqrt{5}) \cup (-2, -\sqrt{3}) \cup (\sqrt{3}, 2) \cup (\sqrt{5}, \infty)$
D.
$(-\infty, -\sqrt{5}) \cup (-2, -\sqrt{3}) \cup (0, \sqrt{3}) \cup (2, \sqrt{5})$

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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

Increasing/Decreasing Strictness|Logarithmic Differentiation Domain|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly differentiate logarithmic functions involving absolute values and perform complex sign scheme analysis.

The primary trap lies in the derivative of $\ln|x^2 - 4|$ .

Students frequently make the mistake of writing it as $\frac{2x}{|x^2 - 4|}$ instead of the correct $\frac{2x}{x^2 - 4}$ .

This error eliminates the sign change contributed by the denominator $x^2 - 4$ , leading to incorrect intervals of monotonicity (Option B).

Additionally, it tests the careful partitioning of the domain around roots, asymptotes, and points where the log argument crosses 1.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Increasing/Decreasing Strictness
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free