Before You Begin

Question #40379

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \setminus \{0\} \to \mathbb{R}$ be an odd function satisfying the differential equation $x^2 f'(x) + 3x f(x) = 4x^2 + 2$ for all $x > 0$ , with the condition $f(2) = 2$ . Let $y = L(x)$ be the oblique asymptote of the curve $y = f(x)$ . If the curve intersects its asymptote at two distinct points with $x$ -coordinates $\alpha$ and $\beta$ ( $\alpha < \beta$ ), let $k$ be the number of points $c \in (\alpha, \beta)$ such that the tangent to $y = f(x)$ at $x = c$ is parallel to $y = L(x)$ . Find the value of $k$ .

Options

A.
0
B.
1
C.
2
D.
3

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

Rational Function Asymptotes|Implicit to Explicit Conversion|Rolle's Theorem Anchors

Why This Question Is Tricky

Common Mistakes

This question tests the applicability conditions of Rolle's Theorem in the context of rational functions and their asymptotes.

Students are baited into defining a difference function $g(x) = f(x) - L(x)$ and observing that $g(-2) = g(2) = 0$ .

A blind application of Rolle's Theorem would lead them to incorrectly conclude that $g'(c) = 0$ for some $c \in (-2, 2)$ , implying at least one point where the tangent is parallel to the asymptote.

However, the lethal trap is that $f(x)$ has a vertical asymptote at $x = 0$ , breaking the continuity requirement of Rolle's Theorem on the interval $[-2, 2]$ .

Consequently, the derivative condition is never met within the interval, and the actual number of such points is zero.

Quick Facts About This Question

Question Information

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