. . Absolute maxima and minima The absolute maxima and absolute minima are referred to describing the largest and smallest values of a function on an interval. Definition .
Let x be a number in the domain D of a function f x ( ) . Then f x is the absolute maximum value of f x ( ) on D , if f x D ) ≥ ( ) ∀∈ and f x is the absolute minimum value of f x ( ) on D if f x D ) ≤ ( ) ∀∈ In general, there is no guarantee that a function will actually have an absolute maximum or absolute minimum on a given interval. The following figures show that a continuous function may or may not have absolute maxima or minima on an infinite interval or on a finite open interval. However, the following theorem shows that a continuous function must have both an absolute maximum and an absolute minimum on every closed interval.
( ) has an absolute minimum but no absolute maximum on −∞∞ ( ) has no absolute extrema on −∞∞ ( ) has an absolute maximum and an absolute minimum on −∞∞ ( ) has no absolute extrema on a b ) . ( ) has an absolute maximum and an absolute minimum on a,b