Optimization — Find the Best Value

"Find the dimensions that maximize/minimize..."

Method:

1. Draw a picture. Identify what you're optimizing.

2. Write a function for the quantity to optimize.

3. Use constraints to reduce to one variable.

4. Take the derivative, set equal to zero, solve.

5. Verify it's a max/min (second derivative test or check endpoints).

Classic problem: Maximize the area of a rectangle with perimeter 100.

Constraint: 2L + 2W = 100, so W = 50 - L.

Optimize: A = L · W = L(50 - L) = 50L - L²

A' = 50 - 2L = 0 → L = 25, W = 25. Square.

Common types: Fencing problems, box construction (cut corners from cardboard), minimum distance, maximum volume. Know the geometry formulas cold.

Reference:

Wikipedia: Optimization

https://en.wikipedia.org/wiki/Mathematical_optimization