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Question #40386

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [0, 2] \to [0, 4]$ be defined by $f(x) = x^3 - 3ax^2 + 3(a^2-1)x + b$ . The number of real pairs $(a, b)$ such that $f$ is invertible is

Options

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

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

Codomain vs Range Distinction|Inverse Existence Criterion|Differentiability vs Continuity

Why This Question Is Tricky

Common Mistakes

This question tests the inverse existence criterion by artificially restricting the codomain.

The trap lies in assuming that matching the boundary values of the domain to the codomain ( $f(0)=0, f(2)=4$ or $f(0)=4, f(2)=0$ ) is sufficient for bijectivity.

Students often solve the boundary equations and find 3 pairs of $(a, b)$ .

However, they fail to check if the derivative's sign change forces the range to expand beyond the codomain, breaking injectivity.

For the increasing case, the roots of $f'(x)$ fall inside $(0, 2)$ , creating local extrema that violate the strict monotonicity required for an inverse to exist.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Codomain vs Range Distinction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free