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Question #40419

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \cos x |\cos x| - \sin x |\sin x|$ and $F(x) = \int_0^x f(t) dt$ . If $M$ and $m$ are the global maximum and global minimum values of $F(x)$ on the interval $[0, 10\pi]$ , find the value of $M - m$ .

Options

A.
$\frac{\pi+2}{2}$
B.
$\frac{\pi+1}{2}$
C.
1
D.
$\frac{\pi}{2}$

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

Allied Angle Periodicity|Fundamental Theorem of Calculus|Global Extrema on Intervals

Why This Question Is Tricky

Common Mistakes

The trap exploits the Fundamental Theorem of Calculus combined with piecewise trigonometric periodicity.

Students often incorrectly assume that for a periodic integrand with $\int_0^{2\pi} f(t) dt = 0$ , the global extrema of the integral $F(x)$ must occur at the quadrant boundaries (yielding a difference of $\pi/2$ ).

Alternatively, they blindly drop the absolute values, reducing the integrand to $\cos 2t$ everywhere (yielding a difference of 1).

The lethal collision occurs because the true global extrema are hidden strictly inside the first and third quadrants, requiring rigorous interval analysis rather than rote symmetry application.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Allied Angle Periodicity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free