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Question #40338

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a real-valued function defined on its maximal domain $D \subset \mathbb{R}$ such that $f$ is continuous on $D$ , differentiable on the interior of $D$ , and satisfies the equation $y^3 - x^3 = 3(y - x)$ for all $x \in D$ . If $f(0) = \sqrt{3}$ , let $I \subseteq D$ be the largest interval containing $x = 0$ such that the restriction of $f$ to $I$ is a one-to-one function. What is the length of the interval $I$ ?

Options

A.
1
B.
2
C.
3
D.
4

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

Function Definition (The 'One-to-One' Constraint)|Variable Separable Constraints|Implicit Differentiation Implicit Domain

Why This Question Is Tricky

Common Mistakes

This question tests the implicit domain of a curve and the identification of true critical points.

The primary trap lies in differentiating the original equation to get $y' = \frac{x^2 - 1}{y^2 - 1}$ .

Students will set the numerator to zero, finding $x = \pm 1$ , and incorrectly assume $x=1$ is a turning point.

This leads them to believe the monotonic interval containing $x=0$ is $[-1, 1]$ , yielding a length of 2.

They fail to realize that at $(1,1)$ , the derivative is a $\frac{0}{0}$ indeterminate form, and on the specific continuous branch passing through $(0, \sqrt{3})$ , the derivative at $x=1$ is actually $-1$ .

Furthermore, it tests the understanding that switching branches at the intersection $(1,1)$ would violate the differentiability constraint.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Function Definition (The 'One-to-One' Constraint)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free