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 de?nition.
De?nition: If there exists an open interval ًa; bق containing c such that f ًxق < f ًcق for all x other than c
in the interval, then f ًcق is a relative maximum of f . If f ًxق > f ًcق for all x in ًa; bق other than c, then
f ًcق 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 de?ned.
De?nition: If c is in the domain of f and for all x in the domain of the function f ًxق @ f ًcق, then f ًcق is
an absolute maximum of the function f . If for all x in the domain f ًxق A fًcق then f ًcق is an absolute
minimum of f . (See Fig. 3-3.)
Note: If de?ned on closed intervals the strictly increasing and decreasing functions possess absolute
extrema
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .