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Question #41592

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: (0, 6) \setminus \{3\} \to \mathbb{R}$ be a differentiable function satisfying the differential equation $(x-3) \frac{dy}{dx} + (x^2 - 6x + 9) y = 0$ with the initial condition $f(2) = e^4$ . Let $A$ be the number of points of inflection of $f(x)$ in its domain. Let $B = \lim_{x \to 3} \ln(f(x))$ . Let $C = 1$ if $f(x)$ attains a global maximum on the interval $[1, 5] \setminus \{3\}$ , and $C = 0$ otherwise. Find the value of $2A + 2B + C$ .

Options

A.
12
B.
13
C.
14
D.
15

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

Variable Separable Constraints|Concavity Analysis|Limit Existence at Discontinuity|Global Extrema on Intervals

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to rigorously analyze a transcendental function derived from a variable separable differential equation with a removable singularity.

The primary trap lies in the evaluation of the global maximum (Trap 266).

Students often blindly find the critical point ( $x=3$ ), calculate the function's limit, and incorrectly assume this supremum is the attained global maximum, leading them to erroneously set $C=1$ .

However, because $x=3$ is explicitly excluded from the domain (Trap 243), the maximum does not exist.

Furthermore, the concavity analysis requires correctly applying the product rule to $f'(x)$ to find the true inflection points at $x=2$ and $x=4$ , bypassing the non-existent critical point.

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