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Question #41360

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function satisfying the differential equation $x f'(x) - 2f(x) = 2x^4$ for all $x \in \mathbb{R}$ . It is given that the application of Lagrange's Mean Value Theorem to $f(x)$ on the interval $[-2, 2]$ yields exactly three distinct points in the open interval $(-2, 2)$ where the tangent to the curve is parallel to the secant line joining the endpoints. If $S$ is the set of all possible integer values of $f''(0)$ , find the sum of all elements in $S$ .

Options

A.
-120
B.
-56
C.
-136
D.
-60

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

LMVT Tangent Parallelism|Linear Differential Equation Structure|General vs Particular Solutions

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge Linear Differential Equations with the geometric conditions of the Mean Value Theorem.

The primary trap lies in the treatment of the integration constant $C$ .

Students often assume $C$ must be determined by a standard Initial Value Problem (IVP) or assume $C$ itself must be an integer (leading to the trap answer -56).

Furthermore, the problem requires enforcing twice-differentiability at $x=0$ to prove the constant $C$ is identical on both sides of the y-axis.

Finally, students must correctly interpret the LMVT condition $f'(c) = m$ as a polynomial equation whose number of roots in a specific domain restricts the continuous range of $C$ , ultimately requiring the summation of $2C$ rather than $C$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	LMVT Tangent Parallelism
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free