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Question #40362

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be the real-valued differentiable solution to the differential equation $\text{sgn}(y) \frac{dy}{dx} = 3 - 2x$ satisfying the initial condition $f(1) = -2$ . If $D = (a, b)$ is the maximum valid domain of $f(x)$ and $m$ is the local minimum value of $f(x)$ on $D$ , find the value of $a + b - 4m$ .

Options

A.
12
B.
15
C.
9
D.
6

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

Signum Function Discontinuity|Implicit to Explicit Conversion|Extraneous Solution Rejection

Why This Question Is Tricky

Common Mistakes

The core trap lies in the implicit to explicit conversion of the differential equation involving the signum function.

Students typically integrate $\text{sgn}(y) dy$ to $|y|$ but often drop the absolute value without considering the domain restriction it imposes.

By blindly writing $y = x^2 - 3x$ , they assume the domain is $\mathbb{R}$ .

However, the absolute value equation $|y| = 3x - x^2$ mandates that $3x - x^2 \ge 0$ .

Furthermore, checking the boundary points $x=0, 3$ reveals that $y=0$ yields $\text{sgn}(0)=0$ , which fails to satisfy the original DE $\text{sgn}(y)y' = 3-2x$ .

This forces the strict domain $(0, 3)$ , testing the student's ability to reject extraneous solutions generated during integration.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Signum Function Discontinuity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free