Finding Minima and Maxima Using Derivatives in Python (NumPy & PyTorch)

Minima and maxima are the lowest and highest points of a function.

• Maxima (Peak) → The function stops increasing and starts decreasing (like the top of a hill).
• Minima (Valley) → The function stops decreasing and starts increasing (like the bottom of a bowl).

👉 To find these points, we check where the derivative is zero (i.e., the function stops changing direction).

Finding Minima and Maxima Using NumPy (Numerical Approach)

We'll use numerical differentiation to find where the function's derivative is close to zero.

import numpy as np

# Define a polynomial function
def polynomial(x):
    return x**3 - 3*x**2 + 2

# Approximate derivative using finite differences
def derivative(f, x, h=1e-5):
    return (f(x + h) - f(x - h)) / (2*h)

# Search for critical points (where the derivative is close to zero)
x_values = np.linspace(-2, 4, 1000)  # Check values in this range
deriv_values = [derivative(polynomial, x) for x in x_values]

# Find where the derivative is close to zero
critical_points = [x for x, d in zip(x_values, deriv_values) if abs(d) < 1e-3]

print("Approximate critical points:", critical_points)

✅ This finds where the function "flattens out," meaning possible minima or maxima.

Finding Minima and Maxima Using PyTorch (Automatic Differentiation)

PyTorch makes this process even easier using autograd.

import torch

# Define x as a trainable variable
x = torch.tensor(2.0, requires_grad=True)

# Define the function
y = x**3 - 3*x**2 + 2  

# Compute derivative
y.backward()  

print("Derivative at x =", x.item(), ":", x.grad.item())

# If x.grad is close to zero, x is a critical point!

✅ PyTorch finds how fast the function is changing at a given point.
If x.grad is near zero, that means we're at a minimum or maximum.

Finding Whether It's a Min or Max

Once we have a critical point, we check if it’s a peak (max) or valley (min) by checking the second derivative:

# Compute second derivative
x.grad = None  # Reset gradient
y2 = (x**3 - 3*x**2 + 2)
y2.backward()  # Compute second derivative

print("Second derivative at x =", x.item(), ":", x.grad.item())

if x.grad.item() > 0:
    print("This is a minimum (valley)!")
elif x.grad.item() < 0:
    print("This is a maximum (peak)!")
else:
    print("This could be a flat point (inflection).")

✅ If the second derivative is positive → Minimum
✅ If the second derivative is negative → Maximum

5️⃣ Summary

• Minima and maxima happen where the derivative is zero (function stops changing).
• NumPy can find approximate critical points numerically.
• PyTorch computes exact derivatives automatically.
• Second derivatives help us check if it's a peak (max) or valley (min).

🎯 Now you can find high and low points of any function using Python! 🚀