Everyday usage and use in mathematics
The word absolute in everyday English means "total" or "complete." The Oxford English Dictionary provides many definitions for the word absolute [1] such as "Free from dependency, autonomous; not relative" or "Complete, perfect."
For example, an absolute monarch is a ruler who has total control over their country. The term absolute zero refers to the smallest possible temperature and a complete lack of atomic movement.
The word maximum [1] is defined as "The highest possible magnitude or quantity of something which is attained, attainable, or customary; an upper limit of magnitude or quantity." The word minimum [1] is the opposite of maximum. Minimum is defined as "The smallest amount or quantity that is possible, usual, attainable, allowable, etc." Both of these words are common in everyday English.
We say that a function $f(x)$ has an absolute maximum at a point $x_0$ in the domain if $f(x_0) \geq f(y)$ for all $y$ in the domain. The point $x_0$ is called an absolute maximum because there is no point anywhere in the domain where $f(y)$ is larger than $f(x_0)$.
Similarly, we say that a function $f(x)$ has an absolute minimum at a point $x_0$ in the domain if $f(x_0) \leq f(y)$ for all $y$ in the domain. The point $x_0$ is called an absolute minimum because there is no point anywhere in the domain where $f(y)$ is smaller than $f(x_0)$.
An absolute extremum can refer to either an absolute maximum or absolute minimum.
The terms absolute maximum and absolute minimum are not frequently used in everyday English, but an everyday English speaker would likely understand what the terms mean because maximum and minimum are everyday english words. The word absolute signifies that there is no way to choose $y$ to make $f(y)$ a larger or smaller value. The term extremum is not used in everyday English, but the related word extreme is. The word extreme means to the greatest possible extent. For example, both Antarctica and Death Valley are known for their extreme temperatures.
Sometimes the terms global maximum and global minimum are used instead of absolute maximum and absolute minimum. These terms have the same meaning and are somewhat more common in higher mathematics. The word "global" emphasizes that the maximum or mimimum is over the entire domain.
Note that the plural of maximum is maxima (not "maximums") and the plural of minimum is minima (not "minimums"). These are irregular words that retain their latin pluralization.
Summary
An absolute maximum (or absolute minimum) is a point in the domain at which a function takes its largest (or smallest) value. The word absolute indicates that there is no point anywhere in the domain at which the function is larger (or smaller).
Importance in mathematics
Finding an absolute maximum or absolute minimum is a very important type of mathematical problem called an optimization problem. Optimization problems are ubiquitous in engineering, science, and business. For example, a company may be interested in finding a delivery route that gives the absolute minimum delivery time. A factory supervisor might be interested in a production plan that produces theabsolute maximum number of units. Optimization problems are an important subject of study in calculus.
In pre-calculus, you can identify an absolute maximum and minimum from looking at the graph. The absolute maximum occurs at the $x$-coordinate of a point where the graph is as high as possible, and the absolute minimum occurs at the $x$-coordinate of a point where the graph is as low as possible.
A function may obtain an absolute maximum or absolute minimum at more than one point if there is a "tie." This means that if there are several points where a graph is at its highest, then all of those points count as absolute maxima.
Exercises
- If I say that a pizza place has the "absolute worst" pizza, what does the word "absolute" mean?
- Where on Earth does the height of the ground acheve its absolute maximum?
- Give an example of a real-world scenario in which you personally had to find an absolute maximum or minimum.
References
[1] Oxford English Dictionary