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Question #41921

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x)$ be a continuous function on $[0, 3]$ such that $f(0) = 0$ and $f'(x) = \frac{|x-1|}{x-1} \frac{2x}{x^2+1}$ for all $x \in (0, 3) \setminus \{1\}$ . Let $S$ be the set of all $c \in (0, 3)$ that satisfy the equation $f'(c) = \frac{f(3) - f(0)}{3}$ . Find the number of elements in $S$ .

Options

A.
0
B.
1
C.
2
D.
3

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

Logarithmic Integration Modulus|Mean Value Theorem Applicability|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the applicability of the Mean Value Theorem (MVT) in the presence of a modulus-induced cusp, combined with logarithmic integration.

Students often blindly integrate the expression by ignoring the modulus sign dependency, incorrectly finding $f(3) = \ln 10$ and subsequently finding 2 false roots for $c$ .

Furthermore, even if integrated correctly, students might assume MVT guarantees a solution $c$ .

However, because the modulus creates a jump discontinuity in the derivative at $x=1$ , $f(x)$ is non-differentiable there, violating MVT's hypothesis.

This creates a 'gap' in the range of the derivative, and the secant slope falls exactly within this gap, resulting in zero solutions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Logarithmic Integration Modulus
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free