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Question #41333

Difficulty: 8/10 | 🟢 Free

The Question

Let $H$ be the right branch of the hyperbola $x^2 - y^2 = 1$ . For a point $P(x,y)$ on $H$ with $y \ge 0$ , let $d(x)$ be the perpendicular distance from $P$ to the asymptote $y=x$ . Define the function $f(x) = d(x) \ln x$ for $x \ge 1$ . Let $S$ be the set of points $x \in (1, \infty)$ where the tangent to the curve $y = f(x)$ is horizontal. Which of the following statements is TRUE?

Options

A.
$S$ is empty because $\lim_{x \to \infty} f(x) = \infty$ .
B.
$S$ contains exactly one element $x_0$ , and the normal to $H$ at $(x_0, \sqrt{x_0^2-1})$ has slope $-\ln x_0$ .
C.
$S$ contains exactly one element $x_0$ , and the normal to $H$ at $(x_0, \sqrt{x_0^2-1})$ has slope $\ln x_0$ .
D.
$S$ contains exactly one element $x_0$ , and the tangent to $H$ at $(x_0, \sqrt{x_0^2-1})$ has slope $-\ln x_0$ .

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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

Rolle's Theorem Root Existence|Normal to Conic|Asymptote Geometry

Why This Question Is Tricky

Common Mistakes

This question tests the application of Rolle's Theorem on an infinite interval by analyzing the asymptotic behavior of a hyperbola.

The primary trap is the assumption that $f(x)$ diverges as $x \to \infty$ because $\ln x \to \infty$ , ignoring that the distance to the asymptote $d(x)$ decays as $O(1/x)$ , forcing $\lim_{x \to \infty} f(x) = 0$ .

Students who fail to properly close the interval at infinity will incorrectly conclude $S$ is empty.

Furthermore, it requires precise algebraic manipulation to link the root of the derivative to the geometric properties of the normal to the conic, penalizing students who cannot synthesize calculus with coordinate geometry.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Rolle's Theorem Root Existence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free