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Question #41536

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_0^x |t^2 - 1| dt$ . Find the area of the region bounded by the curve $y = x \lfloor f''(x) \rfloor$ , the x-axis, and the ordinates $x = 0$ and $x = 2$ , where $\lfloor \cdot \rfloor$ denotes the greatest integer function.

Options

A.
$\frac{19}{4}$
B.
$\frac{17}{4}$
C.
3
D.
$\frac{15}{4}$

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

The Sign-Consistency Principle|Concavity Analysis|Step Function Discontinuities|Rate of Change Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the 'Lethal Collision' between Concavity Analysis and Step Function Discontinuities.

The primary trap lies in blindly differentiating the absolute value function without partitioning the domain, leading students to incorrectly assume $f''(x) = 2x$ globally on $(0, 2)$ (yielding the trap answer 17/4).

Furthermore, embedding $f''(x)$ inside a floor function forces a piecewise integration where the 'Rate of Change' jump at $x=1$ creates a singular point.

Students who fail to apply the Sign-Consistency principle will miss the negative concavity on $(0, 1)$ , or they might forget to take the absolute value for the area, yielding the integral value 3.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	The Sign-Consistency Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free