Local Maxima & Minima

Set f'(x) = 0, classify, optimize.

The idea

A local maximum of a function is a point where its value is at least as large as at every nearby point; a local minimum is one where it is at least as small. Finding these points is the heart of optimization: the largest profit, the smallest cost, and the shortest route are all values of some function at a local maximum or minimum.

At an interior maximum or minimum of a differentiable function the curve has stopped rising and not yet started falling, so its tangent is horizontal: $f'(c) = 0.$ A point $c$ where the derivative vanishes is a critical point, and every interior maximum or minimum is one. The search therefore narrows from a whole interval to the solutions of $f'(x) = 0$.

The converse fails: a critical point need not be a maximum or minimum, since a curve can level off for an instant and continue in the same direction. So we classify each critical point by the sign of $f'$ on either side of it: if $f'$ is negative before $c$ and positive after, $c$ is a minimum; positive before and negative after, a maximum.

To solve an optimization problem, write the quantity of interest as a function of one variable, solve $f'(x) = 0$, and classify the critical points found.

Ways to work on it

• Walkthrough. Find and classify a minimum from f'.
• Practice. Minimum value of a quadratic.
• Hardest. An applied max-area optimization problem.

Not sure where to start? Take the ten-question placement test.