Optimization (Calculus)

Optimization is the practice of finding maximum or minimum values of a function. Standard procedure:

1. Set up the function $f(x)$ to maximise/minimise from the problem statement.
2. Differentiate to get $f'(x)$.
3. Find critical points: solve $f'(x) = 0$ (and identify where $f'$ doesn't exist).
4. Classify each: second derivative test ($f''(c) > 0$ → min; $< 0$ → max), or first derivative sign change.
5. Compare with endpoints if on a closed interval (Extreme Value Theorem).

Classic problems: largest rectangle in a circle, cheapest cylindrical can holding a fixed volume, box of maximum volume from a square sheet.

Multi-variable optimization uses the gradient ($\nabla f = \vec{0}$) and the Hessian matrix. Constrained optimization uses Lagrange multipliers. The technique underlies engineering design, economics, and ML training.