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Question #41917

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the solution set of the system of inequalities: $x^2 - 10x + 16 \le 0$ $x^2 - 4x - 32 \le 0$ Consider the function $f: S \to \mathbb{R}$ defined by: $f(x) = \sum_{k=1}^{10} |x - k| \sin\left(\frac{\pi x}{3}\right)$ Let $P$ be the set of all points in the interior of $S$ at which $f(x)$ is non-differentiable, thereby invalidating the conditions for Lagrange's Mean Value Theorem on $S$ . Find the sum of all elements in $P$ .

Options

A.
16
B.
25
C.
15
D.
26

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

Mean Value Theorem Applicability|Sigma Notation Properties|System of Inequalities

Why This Question Is Tricky

Common Mistakes

This question tests the precise conditions for the applicability of Lagrange's Mean Value Theorem (LMVT) alongside the differentiability of functions involving absolute values and sigma notation.

The primary trap lies in the assumption that $f(x) = |x-k|h(x)$ is always non-differentiable at $x=k$ .

Students often fail to check the derivative explicitly, missing that if $h(k) = 0$ , the left and right-hand derivatives equate to zero, making the function differentiable at that specific boundary point.

By restricting the domain using a system of inequalities to $S = [2, 8]$ , the student must identify the interior integer points and evaluate the differentiability condition $\sin(k\pi/3) = 0$ .

Those who blindly assume non-differentiability at all integers will incorrectly sum all interior integers (yielding 25), falling for the primary distractor.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Mean Value Theorem Applicability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free