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Question #40429

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_0^x t^2 [t^2] dt$ , where $[ \cdot ]$ denotes the greatest integer function. Let $g(x) = \int_0^x f''(t) dt$ , where $f''(t)$ is the second derivative of $f(t)$ evaluated at points where $f'(t)$ is differentiable, and $f''(t) = 0$ otherwise. Find the value of $g(10)$ .

Options

A.
4950
B.
10000
C.
9900
D.
5050

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

Greatest Integer Function (GIF)|Leibniz Rule for Differentiation under Integral|Rate of Change Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the Fundamental Theorem of Calculus (FTC) and the Leibniz rule when dealing with discontinuous functions.

The primary trap is the 'Lethal Collision' of blindly integrating a derivative: students often assume $\int_0^x f''(t) dt = f'(x) - f'(0)$ .

However, because the original integrand contains a Greatest Integer Function, $f'(x)$ has jump discontinuities.

FTC requires the antiderivative to be absolutely continuous.

By ignoring the discrete jump points (where the distributional derivative would yield Dirac-delta impulses), students miss the 'lost' area and arrive at the incorrect trap answer of 10000 instead of the true Riemann integral value of 4950.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Greatest Integer Function (GIF)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free