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Question #40522

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: (0, 5) \to \mathbb{R}$ be defined as $f(x) = (x-1)^2 (x-4)^2 \{x\}$ , where $\{x\}$ denotes the fractional part of $x$ . Let $S$ be the set of all points $x \in (0, 5)$ such that $f$ is differentiable at $x$ and $f'(x) = 0$ . Find the sum of all rational elements in $S$ .

Options

A.
$\frac{42}{5}$
B.
5
C.
$\frac{17}{5}$
D.
$\frac{32}{5}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Even Multiplicity Roots|Rolle's Theorem Anchors|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the interaction between even-multiplicity roots and jump discontinuities (fractional part/GIF).

The primary trap is twofold: first, students may incorrectly assume that {x} makes the function non-differentiable at all integers, missing the fact that the even-multiplicity roots $(x-1)^2$ and $(x-4)^2$ 'absorb' the discontinuity, making $f(x)$ differentiable with $f'(x)=0$ at $x=1, 4$ .

Second, students who ignore the fractional part entirely and differentiate $x(x-1)^2(x-4)^2$ will find incorrect rational roots, leading to a trap answer of 5.

The correct application requires piecewise differentiation and verifying roots within their specific intervals.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Even Multiplicity Roots
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free