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.

Absolute minimum of a quartic function 0.3*(x+3)*(x+2)*(x+1)*(x-1) located at (0.326345,-2.07423)

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.

Absolute minimum of a quartic function -0.3*(x+2.5)*(x+1)*(x)*(x-2) located at (1.3,2.38602)

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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