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Question #40441

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \sin([x^2])$ , where $[ \cdot ]$ denotes the greatest integer function. Which of the following statements is TRUE?

Options

A.
$f(x)$ is strictly increasing on $(0, \sqrt{2})$
B.
$f(x)$ is non-decreasing on $[0, 2)$
C.
$f(x)$ is non-increasing on $[\sqrt{2}, \sqrt{6})$
D.
$f(x)$ is strictly decreasing on $(\sqrt{2}, \sqrt{5})$

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

Monotonicity and Boundedness|Increasing/Decreasing Strictness|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

Students often apply the chain rule blindly, assuming $f'(x) = \cos([x^2]) \cdot 2x$ , and analyze the sign of the derivative to determine strict monotonicity.

This ignores that the Greatest Integer Function creates a step function, meaning the derivative is zero almost everywhere and undefined at integers.

Consequently, the function can never be strictly monotonic on an interval.

The problem forces students to abandon calculus-based monotonicity and instead partition the domain to analyze the discrete sequence of values $\sin(n)$ , testing their conceptual understanding of step functions and radian measures.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Monotonicity and Boundedness
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free