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Existence of Maxima, Intermediate Value Property, Differentiabilty 1

الكلية كلية الهندسة     القسم  الهندسة الميكانيكية     المرحلة 1
أستاذ المادة احمد كاظم حسين الحميري       13/07/2018 14:59:34
A real-valued function f defined on a domain X has a global (or absolute) maximum point at x? if f(x?) ? f(x) for all x in X. Similarly, the function has a global (or absolute) minimum point at x? if f(x?) ? f(x) for all x in X. The value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function.

If the domain X is a metric space then f is said to have a local (or relative) maximum point at the point x? if there exists some ? > 0 such that f(x?) ? f(x) for all x in X within distance ? of x?. Similarly, the function has a local minimum point at x? if f(x?) ? f(x) for all x in X within distance ? of x?. A similar definition can be used when X is a topological space, since the definition just given can be rephrased in terms of neighbourhoods.

In both the global and local cases, the concept of a strict extremum can be defined. For example, x? is a strict global maximum point if, for all x in X with x ? x?, we have f(x?) > f(x), and x? is a strict local maximum point if there exists some ? > 0 such that, for all x in X within distance ? of x? with x ? x?, we have f(x?) > f(x). Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points.

A continuous real-valued function with a compact domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed (and bounded) interval of real numbers (see the graph above).

Finding functional maxima and minima
Finding global maxima and minima is the goal of mathematical optimization. If a function is continuous on a closed interval, then by the extreme value theorem global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the largest (or smallest) one.

Local extrema of differentiable functions can be found by Fermat s theorem, which states that they must occur at critical points. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test, second derivative test, or higher-order derivative test, given sufficient differentiability.

For any function that is defined piecewise, one finds a maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is largest (or smallest).

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 .