Lesson 13: Optimization – Finding Maximum & Minimum Values

1. What is Optimization?

Optimization finds maximum or minimum values of a function — very common in real life and ML.

Examples:

• Maximize profit given constraints
• Minimize cost of materials
• Find best learning rate in gradient descent (ML)

Key idea: at max/min points, the derivative (slope) is zero.

2. Steps for Optimization

1. Define the function to optimize (objective function)
2. Express in terms of one variable (if needed)
3. Find critical points: set \(f'(x) = 0\) and solve
4. Use second derivative test or first derivative test to classify:
   • \(f''(x) > 0\) → local minimum
   • \(f''(x) < 0\) → local maximum
   • \(f''(x) = 0\) → inconclusive
5. Check endpoints if domain is closed interval

Summary – Lesson 13

• Critical points: \(f'(x) = 0\) or undefined
• Second derivative test classifies max/min
• Check boundaries on closed intervals

Next: curve sketching using derivatives.