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Question #40946

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: (0, 7) \to \mathbb{R}$ be a function defined as $f(x) = |x^2 - 7x + 6| ([x] - \{x\})$ , where $[x]$ denotes the greatest integer less than or equal to $x$ , and $\{x\}$ denotes the fractional part of $x$ . Let $S$ be the set of all points in $(0, 7)$ where $f(x)$ is not differentiable. Find the value of $\sum_{x \in S} x$ .

Options

A.
19
B.
20
C.
21
D.
27

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

Algebra of Derivatives|Differentiability vs Continuity|Modulus Function Symmetry|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the 'Algebra of Derivatives' rules.

A common trap in JEE Advanced is assuming that the product of a non-differentiable function (like the modulus at its roots) and a discontinuous function (like the greatest integer function) is automatically non-differentiable.

Students often blindly list all integers and roots of the modulus as points of non-differentiability.

However, at $x=1$ , the sign change of the modulus derivative perfectly counteracts the jump discontinuity of the $2[x]-x$ term, resulting in identical left-hand and right-hand derivatives.

By forcing the evaluation of limits from first principles, this problem filters out students who rely on rote theorems instead of fundamental calculus definitions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Algebra of Derivatives
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free