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Question #51970

Difficulty: 8/10 | 🟢 Free

The Question

Let $U$ be the set of all strictly increasing functions on $\mathbb{R}$ . Consider the family of odd polynomials defined by $f(x) = x^{2n+1} - (2n+1) a x^{2n-1} + (2n+1) b x$ where $n \in \mathbb{N}, n \ge 2$ , and $a, b \in \mathbb{R}$ . For a fixed $a > 0$ , which of the following is the necessary and sufficient condition on $b$ such that $f \in U$ ?

Options

A.
$b \ge \frac{2n-1}{n} a \left( \frac{(2n-1)(n-1)}{n} a \right)^{n-1}$
B.
$b > \frac{2n-1}{n} a \left( \frac{(2n-1)(n-1)}{n} a \right)^{n-1}$
C.
$b \ge \frac{2n-1}{n-1} a \left( \frac{(2n-1)n}{n-1} a \right)^{n-1}$
D.
$b > \frac{2n-1}{n-1} a \left( \frac{(2n-1)n}{n-1} a \right)^{n-1}$

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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

Universal Set Relativity|Polynomial Function Parity|Increasing/Decreasing Strictness|Constant Function Logic

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the boundary conditions of monotonic functions within a universal set of strictly increasing functions.

The primary trap lies in the 'Constant Function Logic' and the strictness of inequalities.

Students often incorrectly assume that a strictly increasing function requires $f'(x) > 0$ everywhere, leading them to choose the strict inequality (Option B).

However, $f'(x) \ge 0$ is sufficient as long as $f'(x) = 0$ does not hold over any interval of positive length.

By using purely symbolic parameters, the problem removes arithmetic crutches, forcing a rigorous algebraic derivation of the global minimum and a conceptual verification of discrete critical points.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Universal Set Relativity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free