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Question #41137

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to (0, \infty)$ be a differentiable function satisfying the differential equation $\frac{dy}{dx} + y = y^3 \left( 1 + \frac{3-2x}{4}e^{2x} \right)$ If $f(x)$ is an injective function on $\mathbb{R}$ , then the maximum possible value of $f(0)$ is:

Options

A.
$\frac{2}{3}$
B.
$\frac{3}{2}$
C.
$\frac{4}{9}$
D.
$\frac{1}{3}$

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

Composite Injectivity/Surjectivity|Bernoulli Equation Transformation|Integration Constant Trap|Second Derivative Test Limitations

Why This Question Is Tricky

Common Mistakes

The problem tests the interplay between Bernoulli ODEs and global monotonicity.

The inner function $v(x) = y^{-2}$ is derived via an integrating factor, introducing a crucial integration constant $C$ that acts as a bifurcation parameter.

The primary trap lies in the monotonicity condition: students often incorrectly assume $v'(x) > 0$ strictly is required for injectivity.

However, when the discriminant $\Delta = 0$ , $v'(x)=0$ yields an inflection point ( $v''(x)=0$ ), which preserves strict monotonicity.

Failing to include the equality $\Delta \le 0$ leads to the false conclusion that no maximum exists (open interval), while ignoring $C$ entirely breaks the domain mapping.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Composite Injectivity/Surjectivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free