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Question #41365

Difficulty: 8/10 | 🟢 Free

The Question

Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $g(x) = (x^2 - 2x + 3)e^{-x}$ . A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined by $f(x) = \int_0^x (g(t) + t g'(t)) dt$ . Which of the following correctly describes the intervals of monotonicity for $f(x)$ ?

Options

A.
Strictly increasing on $(-\infty, 3]$ and strictly decreasing on $[3, \infty)$
B.
Strictly increasing on $(-\infty, 1] \cup [3, \infty)$ and strictly decreasing on $[1, 3]$
C.
Strictly decreasing on $(-\infty, 3]$ and strictly increasing on $[3, \infty)$
D.
Strictly increasing on $[1, 3]$ and strictly decreasing on $(-\infty, 1] \cup [3, \infty)$

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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

Increasing/Decreasing Strictness|Integrating f(x) + xf'(x) forms|Slope Sensitivity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze the strict monotonicity of a function derived from an integral of the form $\int (g(x) + xg'(x)) dx$ .

The primary trap is 'Slope Sensitivity' combined with a repeated root in the derivative.

Students often correctly identify the roots of $f'(x)$ at $x=1$ and $x=3$ , but mistakenly assume the sign of the derivative alternates across all roots (the wavy curve method trap).

By missing the even multiplicity of the root at $x=1$ , they incorrectly conclude the function decreases on $[1, 3]$ , leading to the trap option.

The question rigorously filters for careful sign analysis of $f'(x)$ rather than blind application of alternating signs.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Increasing/Decreasing Strictness
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free