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Question #40360

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \text{sgn}(x^2 - 5x + 6) \cdot |x^2 - ax + b|$ , where $a, b \in \mathbb{R}$ . If Rolle's Theorem is applicable to $f(x)$ on the interval $[1, 4]$ , then the value of $a + b$ is:

Options

A.
10
B.
11
C.
12
D.
13

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

Signum Function Discontinuity|Rolle's Theorem Root Existence|Geometrical Interpretation of Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the student's vigilance regarding the continuity and differentiability prerequisites of Rolle's Theorem, specifically when dealing with piecewise or special functions like signum and modulus.

A common trap is to mechanically check the endpoint condition $f(a) = f(b)$ while ignoring internal discontinuities.

The signum function introduces jump discontinuities where its argument crosses zero.

To satisfy Rolle's Theorem, the function must remain continuous, forcing the accompanying modulus term to evaluate to zero at these exact points, thereby nullifying the discontinuity.

This elegantly links the roots of the inner functions to the theorem's foundational conditions.

Quick Facts About This Question

Question Information

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