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Question #40393

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_0^x \left( \sin\left(4t - \frac{\pi}{6}\right) + \sin\left(2t + \frac{\pi}{6}\right) \right) dt$ . Let $M$ and $m$ denote the number of local maxima and local minima of $f(x)$ in the interval $(0, 2\pi)$ , respectively. Find the value of $10M + m$ .

Options

A.
21
B.
32
C.
43
D.
23

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

ASTC Quadrant Sign Convention|Compound Angle Expansion|First Derivative Test Trap

Why This Question Is Tricky

Common Mistakes

This question weaponizes the First Derivative Test by exploiting students' tendency to assume that the derivative of a periodic trigonometric function alternates signs across all its roots.

By requiring a sum-to-product conversion, the derivative reveals two distinct trigonometric factors.

The trap lies in the overlapping roots ( $2\pi/3$ and $5\pi/3$ ) between $\sin(3x)$ and $\cos(x - \pi/6)$ .

Students who blindly list the critical points and alternate the signs of the derivative will incorrectly classify these points of inflection as local extrema, leading to an overcount of both maxima and minima.

Proper ASTC quadrant analysis or multiplicity checking is required to realize the derivative does not change sign at these double roots.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	ASTC Quadrant Sign Convention
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free