Before You Begin

Question #41401

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \sqrt{|L(x)|} + x$ , where $L(x) = \lim_{t \to 0} \frac{\sin(x^2 t^2) + \cos(2t) - 1}{t^2}$ Let $S$ be the sum of the $x$ -coordinates of all points of local extrema of $f(x)$ . Find the value of $S$ .

Options

A.
$1$
B.
$0$
C.
$\sqrt{2}$
D.
$2$

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

Non-Differentiable Extrema|Algebraic-Trigonometric Graphical Mapping|Indeterminate Forms

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to navigate functions defined via indeterminate limits that map to non-differentiable topologies.

The primary trap lies in the indeterminate form masking the cusps at $x = \pm \sqrt{2}$ .

Students who mechanically apply L'Hôpital's rule or standard derivative tests ( $f'(x) = 0$ ) without analyzing the absolute value's piecewise nature will likely only find the local maximum at $x = 1$ (or fail entirely if they incorrectly solve the positive branch).

By forcing the evaluation of the limit first, the problem filters out candidates who assume continuity implies differentiability, requiring a rigorous check of left/right-hand derivative divergence to identify the non-differentiable extrema.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Non-Differentiable Extrema
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free