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Question #41354

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{\{x\} (x^2 - 5x + 8)}{x}$ for $x \in (0, 5)$ , where $\{x\}$ denotes the fractional part of $x$ . Find the number of solutions to the equation $x f'(x) + f(x) = 0$ in the interval $(0, 5)$ .

Options

A.
0
B.
3
C.
4
D.
5

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Rolle's Theorem Anchors|Integrating f(x) + xf'(x) forms|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the student's mathematical maturity in applying Rolle's Theorem.

The function is designed in the standard product rule derivative form d/dx[x f(x)], which strongly baits students into integrating the equation and finding roots of x f(x) = 0.

Because x f(x) = 0 at all integers in the interval, a rote application of Rolle's Theorem falsely predicts 5 roots.

However, the presence of the fractional part function introduces jump discontinuities at every integer, violating the continuity condition of Rolle's Theorem.

Manually checking the piecewise derivatives reveals that no actual roots exist, trapping students who fail to verify boundary conditions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Rolle's Theorem Anchors
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free