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Question #52000

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \{x \in \mathbb{R} : |x - 2| \le 1\}$ , $B = \{x \in \mathbb{R} : |x - 3| \le 1\}$ , and $C = \{x \in \mathbb{R} : e^x > x^3\}$ . Let $f: A \cup B \to \mathbb{R}$ be defined as: $f(x) = \begin{cases} x^2 - 4x + 5 & \text{if } x \in A \setminus B \\ -x^2 + 6x - 7 & \text{if } x \in A \cap B \\ x^2 - 8x + 17 & \text{if } x \in B \setminus A \end{cases}$ Let $S$ be the sum of the $x$ -coordinates of all points of local extrema of $f(x)$ in the interior of its domain. Find the value of $S \int_1^3 f(x) dx$ .

Options

A.
15
B.
9
C.
6
D.
30

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

Examiner Levers

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the intersection of set theory (Venn diagram regions) with real analysis, specifically targeting the 'Non-Differentiable Extrema' trap.

Students must map the set differences and intersections to piecewise intervals.

The critical trap lies at the boundaries $x=2$ and $x=3$ .

Students relying solely on Fermat's Theorem ( $f'(x)=0$ ) will find no valid extrema in the open intervals and incorrectly conclude $S=0$ or miss one of the points.

Recognizing that local extrema can occur at sharp corners (where LHD $\neq$ RHD) is essential to finding the correct sum $S=5$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Venn Diagram Overlap Logic
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free