Minimum and Maximum

The value $f(c)$ is a relative minimum, or local minimum, of a function if $f(c)$ is the least output of $f$ around $x=c.$ Likewise, the value $f(d)$ is a relative maximum, or local maximum, of a function if $f(d)$ is the greatest output of $f$ around $x=d.$

If the function is continuous, the function switches from increasing to decreasing at a relative maximum or from decreasing to increasing at a relative minimum. Sometimes the phrase relative extrema is used to refer to both relative maximums and relative minimums. Note that a function can have one or more relative extrema, or none at all.

The absolute minimum, or global minimum, of a function is the least output in its whole domain.

The absolute maximum, or global maximum, of a function is defined in a similar way. It is the greatest output of the function in its whole domain.

The absolute maximum of a function is also a relative maximum, and the absolute minimum is also a relative minimum. If a function increases indefinitely, it does not have an absolute maximum. Likewise, if a function decreases indefinitely, it does not have an absolute minimum. The function might still have relative extrema.