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Question #41915

Difficulty: 8/10 | 🟢 Free

The Question

Consider the function $f(x) = \int_{x}^{x^2} t[t] \, dt$ , where $[ \cdot ]$ denotes the greatest integer function. A student attempts to apply the Lagrange Mean Value Theorem to $f(x)$ on the interval $[\frac{1}{2}, 2]$ but realizes it fails due to non-differentiability at a set of points $S \subset (\frac{1}{2}, 2)$ . The sum of the squares of the elements in $S$ is:

Options

A.
3
B.
5
C.
6
D.
8

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|Newton-Leibniz Formula with variable limits|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the applicability of the Mean Value Theorem on functions defined by integrals with variable limits containing jump discontinuities (like the Greatest Integer Function).

A common trap is blindly applying the Newton-Leibniz formula to find $f'(x)$ without realizing that the derivative is undefined at points where the limits of integration hit the discontinuities of the integrand.

Students must identify that while the integral of a piecewise continuous function is continuous, it is non-differentiable at these boundary-crossing points, thus violating MVT conditions.

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