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Question #40796

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all continuous functions $f: [1, 3] \to \mathbb{R}$ satisfying $f(x) \ge \left| \int_{x}^{x^2-4x+5} (t-2) \, dt \right|$ for all $x \in [1, 3]$ . The minimum possible value of the definite integral $\int_{1}^{3} f(x) \, dx$ is:

Options

A.
$\frac{1}{5}$
B.
$\frac{2}{5}$
C.
$\frac{3}{5}$
D.
$\frac{4}{5}$

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

Parabola Shifted Vertex|Substitution (Change of Variable)|Compound Function Boundaries|Inequalities in Definite Integrals

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation of definite integrals involving absolute values and compound parabolic boundaries, a common setup for integral inequalities.

The trap lies in the substitution $u = x-2$ , which shifts the coordinate system.

Students frequently fail to identify the internal roots of the resulting polynomial $u^4 - 3u^2 + 1$ within the integration interval $[0, 1]$ .

By ignoring the sign change at $u = (\sqrt{5}-1)/2$ , they incorrectly compute the net area (yielding $1/5$ ) instead of the absolute area required by the inequality bound, missing the correct answer of $3/5$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Parabola Shifted Vertex
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free