Reference

Minimum and Maximum

Concept

Relative 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.

Concept

Absolute Minimum and Maximum

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.

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