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Question #41336

Difficulty: 8/10 | 🟢 Free

The Question

A curve $C$ is defined parametrically by $x(t) = t^3 + t$ $y(t) = -2t^3 + 9t^2 - 12t + 7[t]$ for $t \in [0, 4]$ , where $[ \cdot ]$ denotes the greatest integer function. What is the absolute maximum value of $y$ on the curve $C$ ?

Options

A.
10
B.
12
C.
14
D.
16

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

Step Function Discontinuities|Parametric form pitfalls|Non-Differentiable Extrema

Why This Question Is Tricky

Common Mistakes

This question exploits the 'Stationary Point Bias'.

Students reflexively rely on Fermat's Theorem, setting $dy/dt = 0$ (or $f'(t)=0$ for the continuous part) to find stationary points.

They will correctly identify $t=2$ as a local maximum of the continuous cubic part and calculate $y(2) = 10$ .

However, they fail to account for the non-differentiable extrema introduced by the step function $[t]$ .

The jump discontinuity at $t=3$ provides a $+7$ boost, which overpowers the $-5$ decrease in the continuous cubic part between $t=2$ and $t=3$ .

This tests the student's ability to recognize that global extrema in piecewise/step functions often occur at points of non-differentiability (jump points), rendering standard derivative tests insufficient.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Step Function Discontinuities
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free