A maximum (plural maxima), in the context of functions, is the largest value of the function either within a given interval, or over the entire domain of the function. In other words, a point on a function is a maximum if its height is greater than or equal to any other point within the given interval.

Point a in the figure above shows the largest value of f(x). In this graph, a is an absolute maximum.

Local and absolute maxima

A local maximum, also referred to as a relative maximum, is a point a within a given interval of a function that satisfies the condition of being the largest value within the given interval. Using typical function notation:

f(a) ≥ f(x) for all x in the given interval

In contrast, an absolute maximum, also referred to as a global maximum, is the largest value of f(x) across its entire domain:

f(a) ≥ f(x) for all x∈ℝ

The figure above shows one local maximum. The absolute maximum of the function is ∞, which f(x) achieves at both ends of its domain (-∞, ∞).