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Question #41185

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \sin^{-1}(\sin x) \cdot \cos^{-1}(\cos x)$ . Find the number of points of inflection of the curve $y = f(x)$ in the interval $(-2\pi, 2\pi)$ .

Options

A.
3
B.
4
C.
7
D.
8

Check Your Answer

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

Inverse Function Periodicity|Geometry Constraint Modeling|Point of Inflection Conditions

Why This Question Is Tricky

Common Mistakes

Students incorrectly assume that any point where the second derivative changes sign (or concavity changes) is automatically a point of inflection.

They find 7 points where $\sin 2x$ changes sign and choose 7.

This question exploits the 'Domain Restriction Trap' of periodic inverse trigonometric functions.

An inflection point strictly requires the existence of a unique tangent line.

The trap tests the product rule on piecewise functions, specifically how a zero of one function ( $u(x)=0$ ) can 'smooth out' the non-differentiable corner of another, leaving exactly 3 valid inflection points out of the 7 candidates.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Inverse Function Periodicity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free