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Question #41594

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to (0, \infty)$ be a differentiable function satisfying $f(0) = 1$ and $f'(x) = f(x) \left( 1 + \frac{2x}{x^2+1} \right) + \frac{(x^2+1)^2}{f(x)} - 2(x^2+1)$ for all $x \in \mathbb{R}$ . Find the value of $f(3)$ .

Options

A.
8
B.
9
C.
10
D.
12

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

Variable Separable Constraints|Arithmetic Mean Inequality|Constant Function Logic|Identity Function Behavior

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to identify singular solutions in non-linear differential equations.

The standard algorithmic approach of separating variables requires dividing by $(y-1)^2$ , which implicitly assumes $y \neq 1$ .

Students who blindly integrate will obtain $\ln|y-1| - \frac{1}{y-1} = x + C$ .

Applying the initial condition $y(0)=1$ yields an undefined LHS, leading many to incorrectly conclude that no solution exists or to make algebraic errors forcing a fit.

The trap is failing to recognize that the constant function $y=1$ (derived via the AM-GM inequality structure $y + 1/y \ge 2$ ) is a valid singular solution that perfectly satisfies both the ODE and the boundary condition, yielding $f(x) = x^2+1$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Variable Separable Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free