Before You Begin

Question #40305

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \{x \in \mathbb{R} : x^2 - 4x + 3 \le 0\}$ and $B = \{x \in \mathbb{R} : x^2 - 6x + 8 \le 0\}$ . A function $g: A \cup B \to \mathbb{R}$ is defined as: $g(x) = \begin{cases} x^2 - 5 & \text{if } x \in A \setminus B \\ 2x - 5 & \text{if } x \in A \cap B \\ 2x - 7 & \text{if } x \in B \setminus A \end{cases}$ Let $f(x) = \int_{2}^{x} g(t) \, dt$ . If $S$ is the set of all interior points in the domain of $f$ where $f(x)$ attains a local extremum, then the sum of all elements in $S$ is:

Options

A.
$6$
B.
$8$
C.
$9$
D.
$11$

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

Venn Diagram Overlap Logic|Non-Differentiable Extrema|Right-Hand vs Left-Hand Derivatives

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to map set-theoretic operations (Venn diagram regions) to piecewise analytical functions and identify non-differentiable extrema.

The primary trap lies at the intersection boundary $x=3$ .

Students commonly rely solely on Fermat's Theorem ( $f'(x)=0$ ) to find local extrema, which correctly identifies the differentiable minima at $x=2.5$ and $x=3.5$ .

However, they frequently overlook the sharp turn (cusp) at $x=3$ where the left-hand derivative is positive and the right-hand derivative is negative, creating a non-differentiable local maximum.

Missing this boundary extremum leads directly to the trap option of 6.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Venn Diagram Overlap Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free