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Question #41397

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function satisfying the differential equation $\frac{dy}{dx} + \text{sgn}(x-1) y = 2 - x$ for all $x \neq 1$ , with the initial condition $f(0) = 0$ . If $x_0$ is the point where $f(x)$ attains its global maximum, then the value of $f(x_0) + x_0$ is:

Options

A.
$e + 1$
B.
$e - 1$
C.
$2e - 1$
D.
$e + 2$

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

Non-Differentiable Extrema|Linear Differential Equation Structure|Integrating Factor (IF) Sign Sensitivity|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This problem tests the interaction between piecewise differential equations and non-differentiable extrema.

The presence of the signum function in the integrating factor forces the solution to be defined piecewise.

Students often mechanically set $f'(x) = 0$ to find extrema.

However, in this function, $f'(x) = 0$ yields no real solutions on either side of the domain.

The trap lies in assuming that extrema only occur where the derivative vanishes, causing students to overlook the critical point at $x=1$ where the function is continuous but non-differentiable (a sharp peak).

Recognizing that $f'(1^-) > 0$ and $f'(1^+) < 0$ is essential to identifying the global maximum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Non-Differentiable Extrema
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free