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Question #41539

Difficulty: 8/10 | 🟢 Free

The Question

Let $F(x) = \int_{0}^{x} \left( \int_{0}^{y} \text{sgn}(y - 2t) e^{t^2} dt \right) dy$ for $x > 0$ . Find the value of $F''(2)$ .

Options

A.
$e - e^4$
B.
$-e^4$
C.
$\frac{e}{2} - e^4$
D.
$e + e^4$

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

Axis of Integration Selection|Newton-Leibniz Formula with variable limits|Second Order Derivative Continuity|Limit Existence at Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Newton-Leibniz formula across a point of discontinuity.

Students often blindly apply the Leibniz rule $\frac{d}{dx} \int_{0}^{x} g(x,t) dt = g(x,x) + \int_{0}^{x} \frac{\partial g}{\partial x} dt$ , which assumes $g(x,t)$ is continuous.

Here, $\text{sgn}(y-2t)$ introduces a jump discontinuity at $t=y/2$ .

The trap is ignoring this discontinuity, leading to the incorrect derivative $-e^{x^2}$ .

The rigorous approach requires partitioning the integral at the discontinuity before differentiating, effectively selecting a valid domain for the fundamental theorem of calculus.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Axis of Integration Selection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free