Point of absolute extremum

Definition

A point of absolute extremum refers to a point in the domain of a function that is either a point of absolute maximum or a point of absolute minimum. Both these are defined below.

Point of absolute maximum

A point $c$ in the domain of a function $f$ is defined as a point of absolute maximum if $f(x)\leq f(c)$ for all $x$ in the domain of the function.

The value $f(c)$ is termed the absolute maximum value.

Note that for a given function, it is possible for there to be more than one point of absolute maximum, but the absolute maximum value must be the same at all points of absolute maximum.

Point of absolute minimum

A point $c$ in the domain of a function $f$ is defined as a point of absolute minimum if $f(x)\geq f(c)$ for all $x$ in the domain of the function.

The value $f(c)$ is termed the absolute minimum value.

Note that for a given function, it is possible for there to be more than one point of absolute minimum, but the absolute minimum value must be the same at all points of absolute minimum.

Relation with point of local extremum

Absolute extremum implies local extremum or endpoint extremum for functions on intervals

The following are true:

• Any point of absolute maximum that is in the interior of the domain of the function is a point of local maximum. Similarly, any point of absolute minimum that is in the interior of the domain of the function is a point of local minimum.
• If a point of absoute maximum for a function of one variable is at an endpoint of the domain with the function defined on one side, it is a point of endpoint maximum, which means that it is a point of one-sided local maximum. For instance, if the absolute maximum occurs at a right endpoint, then that point is a point of local maximum from the left. Similarly, if the absolute maximum occurs at a left endpoint, then that point is a point of local maximum from the right. Similarly for minima.
• For a function whose domain is an interval, any point of absolute maximum must be either a point of local maximum or a point of endpoint maximum. Similarly, any point of absolute minimum must be either a point of local minimum or a point of endpoint minimum.

Existence of absolute extrema

Consider a continuous function $f$ on a closed bounded interval of the form $[a,b]$ with $a<b$. Then, by the extreme value theorem, $f$ attains its absolute maximum value and its absolute minimum value. Thus, $f$ has at least one point of absolute maximum and at least one point of absolute minimum. Further:

• Every point of absolute maximum is either a point of local maximum or a point of endpoint maximum.
• Every point of absolute minimum is either a point of local minimum or a point of endpoint minimum.

Combining the above with the fact that point of local extremum implies critical point, we obtain that:

Every point of absolute extremum is either a critical point or one of the two endpoints $a,b$.