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Question #41150

Difficulty: 8/10 | 🟢 Free

The Question

Let a function $y = f(x)$ be defined parametrically by $x(t) = \tan t$ and $y(t) = [\cos 2t] + \sin^2 2t$ for $t \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$ , where $[ \cdot ]$ denotes the greatest integer function. Which of the following statements is correct regarding the applicability of Rolle's Theorem for $f(x)$ on the interval $[-1, 1]$ ?

Options

A.
Rolle's Theorem is applicable and $f'(c) = 0$ at exactly one point $c \in (-1, 1)$ .
B.
Rolle's Theorem is applicable and $f'(c) = 0$ at exactly three points $c \in (-1, 1)$ .
C.
Rolle's Theorem is not applicable because $f(x)$ is discontinuous at $x = 0$ .
D.
Rolle's Theorem is not applicable because $f(-1) \neq f(1)$ .

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Correct Answer

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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 Traps|Rolle's Theorem Root Existence|Parametric Derivative Endpoint Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Parametric Derivative Endpoint Trap' combined with a Greatest Integer Function (GIF) discontinuity.

Students often blindly apply parametric differentiation $\frac{dy}{dx} = \frac{y'(t)}{x'(t)}$ , ignoring the GIF's piecewise nature.

Doing so yields $y'(t) = 2\sin 4t$ , giving $f'(0) = 0$ , which tricks them into falsely concluding Rolle's Theorem is applicable and verified at $c=0$ .

They fail to realize that the GIF creates an isolated point discontinuity at $t=0$ (where $\cos 2t$ reaches its local maximum of 1), violating the fundamental continuity prerequisite of Rolle's Theorem.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Greatest Integer Function Traps
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free