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Question #41122

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \left[0, \frac{2}{\pi}\right] \rightarrow \left[0, \frac{2}{\pi} + \frac{4}{\pi^2}\right]$ be defined as: $f(x) = \begin{cases} x + x^2 \sin\left(\frac{1}{x}\right), & x \in \left(0, \frac{2}{\pi}\right] \\ 0, & x = 0 \end{cases}$ Consider the following statements: I. $f$ is a surjective function. II. $f$ is strictly increasing on its domain. III. $f$ is not differentiable at $x=0$ because $\lim_{x \to 0^+} f'(x)$ does not exist.

Which of the following represents the correct truth values of (I, II, III)?

Options

A.
(\text{True}, \text{False}, \text{False})
B.
(\text{True}, \text{True}, \text{False})
C.
(\text{False}, \text{True}, \text{True})
D.
(\text{False}, \text{False}, \text{True})

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

Surjective Mapping (Onto)|Mean Value Theorem Applicability|L'Hopital's Pitfall

Why This Question Is Tricky

Common Mistakes

This question exposes the 'Lethal Collision' between L'Hopital's rule, differentiability, and the Mean Value Theorem.

Students often incorrectly assume that if the limit of the derivative does not exist as x approaches a boundary (a common pitfall when evaluating derivatives of oscillating functions), the function is non-differentiable there.

Furthermore, students frequently misuse MVT by assuming $1 - \cos(1/x) \ge 0$ implies global monotonicity, ignoring the dominant negative contribution of $2x \sin(1/x)$ for specific sequences near zero.

Finally, it tests the conceptual understanding that surjectivity on a closed interval relies on the Intermediate Value Theorem and absolute bounds, not strict monotonicity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Surjective Mapping (Onto)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free