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Question #40519

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a real-valued continuous function on its maximal domain $D$ , satisfying the differential equation $y^3 \frac{dy}{dx} = \text{sgn}(x) (x - \alpha)$ where $\alpha > 0$ is a constant. If $D = [-\beta, 6] \cup [20, \infty)$ for some positive constant $\beta$ , find the value of $f(0)^4 + \alpha + \beta$ .

Options

A.
257
B.
77
C.
263
D.
137

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Wavy Curve (Method of Intervals)|Variable Separable Constraints|Integration Constant Trap|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of variable separable differential equations, piecewise integration, and the Wavy Curve method.

The primary trap is the 'Integration Constant Trap' combined with the signum function's jump discontinuity.

Students often forget that integrating across a piecewise boundary requires independent constants ( $C_1$ and $C_2$ ) that must be explicitly equated using the function's continuity at $x=0$ .

Furthermore, ignoring the $\text{sgn}(x)$ for $x < 0$ leads to an incorrect quadratic inequality, preventing the discovery of $\beta$ .

The problem forces students to reverse-engineer the integration constants and parameters from the Wavy Curve domain intervals using Vieta's formulas.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Wavy Curve (Method of Intervals)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free