Page 51 - Schaum's Outline of Theory and Problems of Advanced Calculus

42 FUNCTIONS, LIMITS, AND CONTINUITY [CHAP. 3

Fig. 3-2

MAXIMA AND MINIMA
The seventeenth-century development of the calculus was strongly motivated by questions concern-
ing extreme values of functions. Of most importance to the calculus and its applications were the
notions of local extrema, called relative maximums and relative minimums.
If the graph of a function were compared to a path over hills and through valleys, the local extrema
would be the high and low points along the way. This intuitive view is given mathematical precision by
the following definition.
Definition: If there exists an open interval ða; bÞ containing c such that f ðxÞ < f ðcÞ for all x other than c
in the interval, then f ðcÞ is a relative maximum of f .If f ðxÞ > f ðcÞ for all x in ða; bÞ other than c, then
f ðcÞ is a relative minimum of f . (See Fig. 3-3.)
Functions may have any number of relative extrema. On the other hand, they may have none, as in
the case of the strictly increasing and decreasing functions previously defined.

Definition: If c is in the domain of f and for all x in the domain of the function f ðxÞ @ f ðcÞ, then f ðcÞ is
an absolute maximum of the function f .If for all x in the domain f ðxÞ A f ðcÞ then f ðcÞ is an absolute
minimum of f . (See Fig. 3-3.)
Note:If defined on closed intervals the strictly increasing and decreasing functions possess absolute
extrema.