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Question #40960

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \int_0^x (t^2 - 6t + 8)(|t-3| - 2) \,dt$ Which of the following represents the complete set of intervals where $f(x)$ is strictly increasing?

Options

A.
$(-\infty, 1] \cup [2, 4] \cup [5, \infty)$
B.
$(-\infty, 1] \cup [5, \infty)$
C.
$[1, 2] \cup [4, 5]$
D.
$[2, 4] \cup [5, \infty)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Right-Hand vs Left-Hand Derivatives|Increasing/Decreasing Strictness|Sign Flipping Rule

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to rigorously evaluate the sign of a derivative composed of multiple factors, specifically targeting the reconciliation of negative regions.

A common trap is assuming that $A \cdot B \ge 0$ implies only $A \ge 0$ and $B \ge 0$ , leading students to completely miss the interval $[2, 4]$ where both factors are negative.

Additionally, students who improperly drop the absolute value by assuming $t \ge 3$ will incorrectly identify the interval as $[2, 4] \cup [5, \infty)$ , failing to account for the left-hand behavior of the non-differentiable term.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Right-Hand vs Left-Hand Derivatives
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free