Optimization

Why optimization matters

Businesses want to maximize profit and minimize cost; engineers want the strongest beam or the largest volume from a fixed sheet of metal. These are optimization problems, and calculus solves them by finding where a model's rate of change is zero. Because a smooth curve turns around exactly where its tangent is horizontal, the best value occurs at a stationary point (or at the boundary of the allowed values).

Finding local maxima and minima

A local maximum is a peak — the function increases up to it and decreases after. A local minimum is a trough. Both are stationary points, so the method is: 1. Differentiate and solve $f'(x) = 0$ to find the $x$-coordinates of stationary points. 2. Classify each as a maximum, minimum, or point of inflection. 3. Substitute back to