Maxima and minima

On an interval I, f has a maximum value f(c) at c if f(c) ≥ f(x) for every x in I, and a minimum value f(c) if f(c) ≤ f(x) for every x in I. Either one is an extreme value, and c is a point of extremum.

Worked example: f(x) = x² has minimum 0 at x = 0 and no maximum on R. On [−2, 1] it has maximum 4 at x = −2.

Worked example: |x| has minimum 0 at 0 and no maximum on R, though on [−2, 1] its maximum is 2.

Worked example: f(x) = x on the open interval (0, 1) has neither: any value can be beaten by one nearer the end, and the ends are not included.

Local maximum at c: f(c) ≥ f(x) for all x in some open interval around c. Local minimum: f(c) ≤ f(x) nearby. A local maximum need not be the highest value overall.

Theorem: if f has a local extremum at an interior point c and is differentiable there, then f′(c) = 0, so the tangent is horizontal.

A point c with f′(c) = 0 or f′(c) undefined is a critical point. Every local extremum is at a critical point, but not every critical point is an extremum.

Worked example: x³ has f′(0) = 0, yet it is below 0 on the left and above 0 on the right, so 0 is a point of inflection, not an extremum.