Everyday usage and use in mathematics
The word local is defined by The Oxford English Dictionary as [1] "Limited, restricted, or peculiar to a particular place or region."
For example, a local telephone call is a call to someone else in the same city or region. A local restaurant is a restaurant that can only be found in a particular city.
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 local maximum at a point $x_0$ in the domain if $f(x_0) \geq f(y)$ for all $y$ in the domain that are sufficiently close to $x_0$. The point $x_0$ is called an local maximum because there is no point in the domain close to $x_0$ where $f(y)$ is larger than $f(x_0)$. Note that there may be points $y$ that are far from $x_0$ where $f(y)$ is larger than $f(x_0)$.
Similarly, we say that a function $f(x)$ has a local minimum at a point $x_0$ in the domain if $f(x_0) \leq f(y)$ for all $y$ in the domain that are sufficiently close to $x_0$. The point $x_0$ is called a local minimum because there is no point close to $x_0$ in the domain where $f(y)$ is smaller than $f(x_0)$. Note that there may be points $y$ thar are far from $x_0$ where $f(y)$ is smaller than $f(x_0)$.
A local extremum can refer to either a local maximum or local minimum.
The terms local maximum and local minimum are not frequently used in everyday English. The word local has a particular meaning in mathematics; a statement that holds ``locally" holds on a small portion of the domain. 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.
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.
A local extremum of a function $f$ is sometimes called a turning point of $f$. This term comes from looking at the graph of $f$--at a turning point, the graph will turn around from going upwards to going downwards, or from going downwards to going upwards.
Sometimes the terms "relative maximum" and "relative minimum" are used instead of "local maximum" and "local minimum."
Summary
An local maximum (or local minimum) is a point in the domain at which a function takes larger (or smaller) values than all nearby points. The word local indicates that there may be points far from the local maximum (or minimum) at which the function is larger (or smaller).
Importance in mathematics
Finding a local maximum or minimum is an important application of calculus. In calculus, it is simple to identify the local maxima and minima of a function. This is sometimes used in the process of finding an absolute maximum or absolute minimum: An absolute maximum or minimum will necessarily be a local maximum or minimum, so an absolute maximum can be found by listing all of the local maxima and choosing the one that gives the largest value when plugged into the function.
In pre-calculus, you can identify a local maximum and minimum from looking at the graph. The absolute maximum occurs at the $x$-coordinate of a point where the graph is at the top of a hill, and an absolute minimum occurs at the $x$-coordinate of a point where the graph at the bottom of a valley.
Exercises
- If I tell you to go to your "local post office," which post office am I telling you to go to?
- Identify a place on Earth other than the peak of mount Everest at which the height of the ground reaches a local maximum.
References
[1] Oxford English Dictionary