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Question #41369

Difficulty: 8/10 | 🟢 Free

The Question

Let a function $f(x)$ be defined as: $f(x) = \lim_{N \to \infty} \sum_{n=1}^{N} (2n-1) \left( \frac{x^2-3x}{x^2+3} \right)^n$ Let $S$ be the set of all critical points of $f(x)$ . Find the sum of the elements of $S$ .

Options

A.
$\frac{13}{4}$
B.
$\frac{1}{4}$
C.
$-\frac{3}{4}$
D.
$1$

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

Critical Point vs Stationary Point|Arithmetic-Geometric Progression (AGP)|Infinite GP Convergence

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of infinite series convergence and differential calculus.

The primary trap is mechanically differentiating the closed-form sum of the AGP without establishing its domain of validity ( $|r(x)| < 1$ ).

Students who ignore the domain constraint will incorrectly include $x = -3$ as a critical point, leading to the trap answer $1/4$ .

Furthermore, it tests the chain rule on a composite series sum, catching students who only solve $r'(x) = 0$ and miss the roots originating from $dS/dr = 0$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Critical Point vs Stationary Point
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free