Peaks, Pits, and Passes: Exploring Maxima, Minima, and Saddle Points with SageMath

🧗‍♀️ Peaks, Pits, and Passes: Finding Maxima, Minima, and Saddle Points with SageMath

Welcome Mathsmagic

🌄 Imagine This:

You’re hiking in the mountains. You reach the summit—no direction leads higher. That’s a maximum. Later, you descend into a valley where every direction goes uphill. That’s a minimum. Then you reach a mountain pass—a saddle point. You're high in one direction, but low in another. It’s a critical junction where the landscape twists.

In calculus, we use this intuition to analyze surfaces and optimize real-world systems—from minimizing energy usage in drones to maximizing profits in economics.

🧭 Definitions First: Peaks, Pits, and Saddles

Let’s ground ourselves with some terminology:

• Local Maximum: A point where all nearby points have lower function values.
• Local Minimum: A point where all nearby points have higher function values.
• Saddle Point: A point where the function is neither a max nor min, but the gradient still vanishes.

🔍 Connect to Visuals: Think of a landscape.

A peak curves down in all directions. A valley curves up. A saddle curves up in one direction and down in another. We’ll see this twist shortly!

🧪 Visualizing a Saddle in SageMath

Let’s bring this twisty terrain to life:

SageMath code -Plotting the saddle—upward curve

Visualization -Plotting the saddle—upward curve

🪑 Imagine This: Picture a horse’s saddle—upward curve where you sit, downward along the horse’s spine. That’s what this surface resembles.

Your Turn:

Try Changing:the coefficients:

How does the shape stretch or compress?

🧮 Finding Critical Points

So how do we locate these peaks, pits, and passes?

We start by identifying critical points—where the gradient vanishes:

Sage math code -Finding Critical Points

🧠 Imagine This: A ball resting on a hilltop or in a valley has no reason to roll—it’s flat! That’s what a gradient of zero looks like.

🧠 Second Derivative Test: Classifying the Landscape

Once we find flat spots, we ask: What kind of spot is it?

We use the Hessian matrix and its determinant

D = f_xx f_yy- (f_xy)²

Second Derivative Test Sagemath code

🪞 Think of Curvature: The determinant tells us about the overall "shape" of the curvature at the critical point.

Your Turn:

• D>0,f_xx>0 : Local Minimum(Bowl-shaped)
• D>0,f_xx< 0 : Local maximum (dome-shaped)
• D<0 : Saddle point (twisting curvature)

📊 Full Example: Let’s Analyze a Surface



Analyze a Surface Sagemath code

🔍 Visual Clue: Let’s plot it!



Analyze a Surface Sagemath code



Visualizing a Analyze a Surface

Point	fxx	fyy	D	Type
(0,0)	-6	6	-36	Saddle

🧭 Interpretation: The determinant tells us this is a saddle—the curvature bends in opposing directions.

🌍 Real-World Applications

🔧 Engineering Design:

Maximize strength, minimize material usage. Peaks = stress points, pits = stability zones.

🤖 Machine Learning:

Minimize the loss function to optimize model predictions.

📡 Drone Navigation:

Maximize camera coverage while minimizing power use.

💰 Economics:

Maximize profit or minimize cost by analyzing cost/revenue surfaces.

🧬 Biology:

Predict population behavior by studying equilibrium points in dynamic models.

⚛️ Physics:

Particles settle into minima of potential energy—nature’s own optimization.

🧠 Reflect & Explore

🎯 Challenge: Try plotting this function:

Where are the saddle points? How would you classify them?

🔄 Your Turn: Create your own landscape using polynomial functions. Try combining terms like:

• x₂,-y₂ ,x , xy Can you design a surface with:
• A single local max
• Two local mins
• One saddle?

🔍 What's Next?

In our next post, we’ll explore how these ideas evolve in three dimensions—enter gradient descent, contour maps, and optimization with constraints. Get ready to solve real-world challenges using the power of calculus and SageMath!

Stay curious—and keep climbing!

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