Monotonicity and Extrema of a Cubic Function

Overview

This animation analyzes the monotonicity and extreme points of the function $f(x)=x^{3}-3x$. Viewers see the graph, the first‑derivative test, and the second‑derivative test, learning how to locate intervals of increase/decrease and identify local maxima and minima.

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Phases

#	Phase Name	Duration	Description
1	Title Intro	~2s	The title "Monotonicity & Extrema of $f(x)=x^{3}-3x$" fades in at the top.
2	Plot Appearance	~3s	The cubic curve is drawn across the coordinate plane; axes appear with tick marks.
3	Critical Points Reveal	~4s	Points where $f'(x)=0$ ($x=-1,\\;x=1$) appear as highlighted dots; their x‑coordinates slide in from the left.
4	First‑Derivative Test	~6s	The derivative $f'(x)=3x^{2}-3$ is plotted in a faint background; a moving vertical line sweeps across the x‑axis, showing the sign of $f'$ and shading the intervals of increase (green) and decrease (red).
5	Interval Summary	~3s	A concise label "Increasing on $(-\infty,-1)\cup(1,\infty)$" and "Decreasing on $(-1,1)$" fades in at the bottom.
6	Second‑Derivative Test	~6s	The second derivative $f''(x)=6x$ is displayed; arrows point to the critical points, indicating $f''(-1)<0$ (concave down → local maximum) and $f''(1)>0$ (concave up → local minimum).
7	Extrema Highlight	~4s	The local maximum and minimum are emphasized with larger markers and small vertical lines showing the function values $f(-1)=2$ and $f(1)=-2$.
8	Summary Outro	~2s	All auxiliary graphics fade, leaving the original cubic curve and a final caption "Monotonicity & extrema identified" at the bottom.

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Layout

┌─────────────────────────────────────────────┐
│                 TOP AREA (Title)            │
├──────────────────────┬──────────────────────┤
│                      │                      │
│     LEFT AREA        │     RIGHT AREA       │
│  (Graph of f(x))     │  (Derivative plots, │
│                      │   formulas, notes)  │
├──────────────────────┴──────────────────────┤
│                 BOTTOM AREA (intervals,   │
│                 summary captions)          │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "Monotonicity & Extrema of $f(x)=x^{3}-3x$"	Fades in at Phase 1
Left	Coordinate plane with the cubic curve, critical points, and shading for increase/decrease	Primary visual focus
Right	Small inset plots of the first and second derivatives, plus brief formula labels (e.g., $f'(x)=3x^{2}-3$)	Appears starting Phase 4
Bottom	Text labels for increasing/decreasing intervals and final summary caption	Small font, appears in Phases 5 and 8

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Notes

• The entire animation stays within a single Manim Scene class.
• No spoken narration is assumed; all information is conveyed visually.
• All durations are approximate; the total runtime is about 30 seconds, well within the required limit.
• Colors: green for increasing intervals, red for decreasing, blue for the original function, orange for the first derivative, purple for the second derivative.
• Critical points are marked with a solid white dot surrounded by a thin colored halo (green for max, red for min) to draw attention.
• Transitions are simple fades or linear motions; no complex effects are needed.