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Question #40374

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \setminus \{2\} \to \mathbb{R}$ be defined as $f(x) = \lim_{h \to 0} \frac{1}{h} \int_{x}^{x+h} (2t + t^2 \sin^2(t-x)) \, dt + \frac{|x-2|}{x-2} (x^2 - 4x + c)$ Find the minimum integral value of $c$ for which $f(x)$ is an injective (one-to-one) function.

Options

A.
2
B.
3
C.
4
D.
5

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

Function Definition (The 'One-to-One' Constraint)|Indeterminate Forms|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze the injectivity of a piecewise function defined by a modulus and an indeterminate limit form.

A common trap is assuming that if $f'(x) > 0$ on all continuous intervals, the function is globally one-to-one.

However, the modulus term creates a jump discontinuity at $x=2$ .

If the left-hand limit exceeds the right-hand limit, the function 'folds' back on its range, failing the horizontal line test despite being locally monotonic.

Students must rigorously evaluate the limits from both sides of the discontinuity to ensure the ranges do not overlap.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Function Definition (The 'One-to-One' Constraint)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free