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Question #41335

Difficulty: 8/10 | 🟢 Free

The Question

Let $\omega_k = \exp\left(\frac{i 2\pi k}{13}\right)$ for $k=0, 1, \dots, 12$ . Define the function $g: \mathbb{R} \to \mathbb{R}$ as $g(x) = \frac{1}{13} \sum_{k=0}^{12} (x - \omega_k)^{13}$ Let $f(x) = [g(x)] \ln(1 + 3x) + \text{sgn}(g(x)) \sin(2\pi x)$ , where $[ \cdot ]$ denotes the greatest integer function and $\text{sgn}(\cdot)$ denotes the signum function. If $L$ and $R$ are the left-hand and right-hand derivatives of $f(x)$ at $x=0$ respectively, find the value of $L + R$ .

Options

A.
$-9 - 4\pi$
B.
$-6 - 4\pi$
C.
$-3 - 2\pi$
D.
$-12 - 4\pi$

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

Step Function Discontinuities|Roots of Unity Properties|Chain Rule Inner Function Differentiability

Why This Question Is Tricky

Common Mistakes

This question tests the differentiability of composite functions involving step functions and roots of unity.

The core trap lies in the blind application of the chain rule or assuming the step function $[g(x)]$ remains constant in a full two-sided neighborhood of $x=0$ .

Because $g(x) = x^{13} - 1$ crosses an integer at $x=0$ , the greatest integer function jumps from $-2$ to $-1$ .

Students who fail to check the one-sided limits of the inner function will incorrectly assume $[g(x)] = -1$ for all $x$ near $0$ , leading to the trap answer $-6 - 4\pi$ .

This highlights the necessity of evaluating one-sided derivatives independently when the outer function has jump discontinuities.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Step Function Discontinuities
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free