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Question #41469

Difficulty: 8/10 | 🟢 Free

The Question

Let $h(x)$ be the unique real-valued function satisfying the implicit relation $h^3(x) + 3h(x) = 4|x|$ for all $x \in \mathbb{R}$ . Let $f(x)$ be a differentiable function on $\mathbb{R} \setminus \{0\}$ satisfying the differential equation $x f'(x) + f(x) = (3h^2(x) + 3) h'(x) + 4x^3$ If $f(1) = 3$ and $f(-1) = 1$ , find the value of $2f(2) - 2f(-2)$ .

Options

A.
40
B.
44
C.
36
D.
32

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

Integrating f(x) + xf'(x) forms|Implicit Differentiation Rate|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

The question tests the 'Integration Constant Trap' in the context of piecewise or discontinuous functions.

The student is given an implicit relation for $h(x)$ , which upon implicit differentiation yields a signum function.

Substituting this into the differential equation for $f(x)$ results in a discontinuous right-hand side at $x=0$ .

The critical trap is assuming a single constant of integration $C$ globally when integrating $(xf(x))'$ .

Because the domain $\mathbb{R} \setminus \{0\}$ is disconnected, the integration must be split into $x>0$ and $x<0$ with independent constants $C_1$ and $C_2$ .

Failing to do so leads to incorrect trap answers like 44 or 36.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integrating f(x) + xf'(x) forms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free