Find Local Extrema Game/Worksheet

Local Extrema Game/Worksheet
This game generates a random function, asks you to use the First Derivative Test to find the x-coordinates of the local maxima and minima, and checks your answers. Once you submit, it draws the graph to visually confirm your math. Scroll down the page for more details.

 

 

How to Play The Game

1. Analyze the Target Node
   When a problem initializes, the screen will output a continuous polynomial function f(x) inside the main blue console block (e.g., f(x) = x³ - 3x² + 1).
   

2. Formulate and Compute
   Grab a scratchpad and perform your calculus calculations:
   Compute the first derivative: f’(x).
   Solve the equation f’(x) = 0 to extract your critical points.
   Run an interval sign test on either side of those points to determine slope movement.
   

3. Log the Coordinates
   Type your results directly into the input fields:If a critical point represents a Local Maximum, input its value into the top box.
   If it represents a Local Minimum, input its value into the bottom box.
   Formatting: If a function features multiple answers (like a wave tracking several dips), separate the integers with commas (e.g., -1, 1). If no maximum or minimum exists, leave that block completely blank.
   

4. Verify & Visualize
   Click Check Answers. The simulation engine checks your submission against its data table.
   Correct Submission: A green confirmation banner triggers, awarding completion points.
   Incorrect Submission: The interface instantly reveals the true analytical coordinates.
   The Diagnostic Canvas: A real-time graphing window unrolls at the base of the app, charting the function path visually. Red markers denote your local peaks (maxima), while green markers map out the local valleys (minima), providing instant visual validation of your sign charts.
   

The Underlying Mathematics
To win the game consistently, you must master the mechanics of the First Derivative Test. The math inside the game’s code checks your understanding of how analytical slope behavior reveals geometric features.

Step 1: Find the Critical Points
First, find the derivative function, f’(x), which represents the slope of the tangent line at any given coordinate. Critical numbers occur where the derivative is zero or undefined:

f’(x) = 0

For example, given the game’s first problem:

f(x) = x³ - 3x² + 1
f’(x) = 3x² - 6x

Setting it equal to zero and factoring isolates the critical coordinates:

3x(x - 2) = 0 ⇒ x = 0 and x = 2

Step 2: Test the Sign Intervals
The critical numbers divide the domain into distinct intervals. We select a test point within each interval to determine if the slope is positive (increasing) or negative (decreasing):

Interval (-∞, 0): Testing x = -1 ⇒ f’(-1) = 3(-1)² - 6(-1) = 9 > 0 (Increasing)
Interval (0, 2): Testing x = 1 ⇒ f’(1) = 3(1)² - 6(1) = -3 < 0 (Decreasing)
Interval (2, ∞): Testing x = 3 ⇒ f’(3) = 3(3)² - 6(3) = 9 > 0 (Increasing)

Step 3: Classify the Extrema
The game verifies your answers using the core theorem of the First Derivative Test:
Local Maximum: If f’(x) changes from positive to negative as it passes through a critical point c, then f(c) is a local maximum. (The curve climbs up, hits a peak, and turns down).
Local Minimum: If f’(x) changes from negative to positive as it passes through a critical point c, then f(c) is a local minimum. (The curve drops down, bottoms out in a valley, and climbs up).

Therefore, for this problem, x = 0 is a Local Maximum (+ to -) and x = 2 is a Local Minimum (- to +).

Finding Local Maxima and Minima by Differentiation

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