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Question #40520

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a real-valued continuous function defined on $\mathbb{R}$ satisfying the implicit equation $e^y + y^3 = \int_0^x \frac{t^2 - t - 2 + (t^2 - 4)|t - 1|}{t^2 + 1} dt$ If $S$ is the set of all $x \in \mathbb{R}$ for which $f(x)$ is strictly decreasing, then $S$ is equal to:

Options

A.
$[-\sqrt{3}, 2]$
B.
$[-\sqrt{3}, \sqrt{3}] \cup [2, \infty)$
C.
$(-\infty, -1-\sqrt{2}] \cup [-1+\sqrt{2}, 2]$
D.
$[-2, 1] \cup [2, \infty)$

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

Wavy Curve (Method of Intervals)|Implicit Differentiation Rate|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Trap' within the Method of Intervals (Wavy Curve) combined with implicit differentiation.

Students often blindly extract roots from the terms inside and outside the modulus, placing them on a number line to alternate signs.

Here, the implicit differentiation yields a rate of change whose sign depends on a piecewise function.

The trap is that the critical point of the modulus ( $x=1$ ) does not cause a sign change, and the roots of the factors change depending on the domain.

Ignoring the piecewise definition leads to false roots like $\sqrt{3}$ or $-1+\sqrt{2}$ , which are invalid in their respective domains, leading directly to the distractor options.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Wavy Curve (Method of Intervals)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free