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Question #41101

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_0^x (t^2 - 2|t| + 1)^2 (t^2 - 9)^2 dt$ . Let $S = \{x \in \mathbb{R} : f'(x) = 0\}$ . A relation $R$ is defined on $S$ such that for $a, b \in S$ , $a R b$ if and only if $a < b$ and there exists $c \in (a, b)$ satisfying $f''(c) = 0$ . Find the number of elements in $R$ .

Options

A.
3
B.
4
C.
5
D.
6

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Empty and Universal Relation|Rolle's Theorem Root Existence|Chain Rule Inner Function Differentiability|Even Multiplicity Roots

Why This Question Is Tricky

Common Mistakes

This question tests the limits of Rolle's Theorem by introducing a non-differentiable point within the interval.

Students often reflexively assume that if $f'(a) = f'(b) = 0$ , then $f''(c) = 0$ must have a solution in $(a, b)$ .

Here, the 'Even Multiplicity' trap ensures $f'(x)$ doesn't change sign at $x=\pm 1$ , creating a symmetric positive bump on $(-1, 1)$ .

However, the inner absolute value function makes $f'(x)$ non-differentiable at its peak $x=0$ .

Consequently, Rolle's Theorem fails on $[-1, 1]$ , making the relation empty over this specific sub-interval and reducing the total valid pairs from 6 to 5.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Empty and Universal Relation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free