Minimum

A minimum (plural minima), in the context of functions, is the smallest 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 minimum if its height is less than or equal to any other point within the given interval.

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

Local and absolute minima

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

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

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

f(a) ≤ f(x) for all x∈ℝ

The figure above shows a local and an absolute minimum for f(x). Note that while the figure only shows two minima, a function can have any number of minima, not just two.