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Question #41359

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \sum_{k=1}^{8} \frac{x^2}{(k+[x])(k+1+[x])}$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Find the number of points $c \in (1, 3)$ such that the tangent to the curve $y = f(x)$ at $x = c$ is parallel to the chord joining the points $(1, f(1))$ and $(3, f(3))$ .

Options

A.
0
B.
1
C.
2
D.
3

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

LMVT Tangent Parallelism|Method of Differences|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the preconditions of the Lagrange Mean Value Theorem (LMVT).

Students often blindly apply LMVT, assuming that a continuous-looking algebraic expression guarantees at least one point $c$ where the tangent is parallel to the secant.

By embedding a Greatest Integer Function (GIF) inside a Method of Differences series, the function evaluates to a piecewise quadratic that has jump discontinuities at integer values.

Because $f(x)$ is discontinuous at $x=2$ , LMVT is invalid on $[1, 3]$ .

Partitioning the interval and checking the derivative in each continuous segment reveals that no such point $c$ exists, trapping students who guess $\ge 1$ based on theorem memorization.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	LMVT Tangent Parallelism
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free