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Question #41261

Difficulty: 8/10 | 🟢 Free

The Question

Let $P(x) = x^2 - 6x + 8$ . For each $n \in \mathbb{N}$ , let $\alpha_n(x)$ and $\beta_n(x)$ be the roots of the quadratic equation $y^2 - (2n+1)P(x) y + n(n+1)(P(x))^2 = 0$ Define the function $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = \sum_{n=1}^{\infty} \tan^{-1}\left( \frac{\alpha_n(x) + \beta_n(x) - 2n P(x)}{1 + \alpha_n(x) \beta_n(x)} \right)$ Let $M$ be the supremum of $f(x)$ on $\mathbb{R}$ , and $m$ be the value of $f(x)$ at its unique local maximum. Find the value of $M + m$ .

Options

A.
$\frac{\pi}{4}$
B.
$-\frac{\pi}{2}$
C.
$\frac{3\pi}{4}$
D.
0

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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

Infinite Series involving Inverse Tangent|Quadratic Roots in Sequences|Critical Point vs Stationary Point|First Derivative Test Trap

Why This Question Is Tricky

Common Mistakes

This question tests the convergence of a telescoping inverse trigonometric series whose terms depend on a parameter, leading to a piecewise continuous function.

The primary trap lies in blindly applying the First Derivative Test by differentiating the series term-by-term.

Doing so yields $f'(x) = \frac{-P'(x)}{1+P(x)^2}$ , which incorrectly suggests that $f(x)$ is strictly increasing for $x < 3$ and strictly decreasing for $x > 3$ , making $x=3$ appear as the absolute global maximum.

Students falling for this will assume the supremum $M$ and local maximum $m$ are both $-\pi/4$ , leading to the lethal trap answer $-\pi/2$ .

The correct approach requires evaluating the infinite limit first, revealing jump discontinuities at the roots of $P(x)$ and showing that the true supremum is $\pi/2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Infinite Series involving Inverse Tangent
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free