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Question #41918

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x)$ be a continuous function defined on $[-1, 1]$ that satisfies the differential equation $x f'(x) + f(x) = x \ln|x|$ for all $x \in [-1, 1] \setminus \{0\}$ . Let $m$ be the number of points in $(-1, 1)$ where $f(x)$ is non-differentiable. Let $n$ be the number of points $c \in (-1, 1)$ such that $f'(c) = \frac{f(1) - f(-1)}{2}$ . Find the value of $m^2 + n^2$ .

Options

A.
4
B.
5
C.
1
D.
2

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Mean Value Theorem Applicability|Logarithmic Integration Modulus|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the applicability of the Mean Value Theorem (MVT) and the behavior of functions derived from logarithmic integration of a modulus.

Students often blindly assume that a continuous solution to a differential equation is differentiable everywhere.

Here, integrating $(xf(x))' = x\ln|x|$ yields a function with a cusp at $x=0$ , making it non-differentiable ( $m=1$ ).

A common trap is assuming $m=0$ (yielding 4) or assuming that the failure of MVT implies no points satisfy the derivative condition (yielding 1).

The careful student identifies the non-differentiability while still correctly solving the derivative equation to find $n=2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Mean Value Theorem Applicability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free