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Question #41345

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \int_{x}^{2x} t \sin^{-1}(\sin t) \, dt$ Find the value of $f''\left(\frac{5\pi}{6}\right)$ .

Options

A.
$6\pi$
B.
$\frac{35\pi}{3}$
C.
$-\frac{26\pi}{3}$
D.
$\frac{10\pi}{3}$

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

Rate of Change Dependency|Newton-Leibniz Formula with variable limits|Composition Inverse Identity Traps

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Newton-Leibniz formula combined with the piecewise nature of composite inverse trigonometric functions.

The primary trap is the assumption that $\sin^{-1}(\sin t) = t$ globally, which leads to incorrect derivatives (Option B).

Students must identify the correct principal value branches for both $x$ and $2x$ in the neighborhood of the evaluation point before differentiating a second time.

Forgetting the chain rule multiplier in the Leibniz rule is another common pitfall (Option D).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Rate of Change Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free