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[[ملف:Extrema example original.svg|تصغير|280بك|النقاط العظمى والصغرى المحلية والعامة للدالة:cos(3π''x'')/''x'', 0.1≤''x''≤1.1]]
في [[الرياضيات]], النقاط '''العظمى''' و'''الصغرى''', تعرف عموماً '''بالنقاط الحرجة''' هي تلك النقاط التي تكون عندها قيمة [[الدالة]] أعلى مايمكن أو أدنى مايمكن ضمن جوار معرف (منحنى حرج) أو على [[نطاق الدالة]]
بشكل عام، تعرف النقاط العظمى والصغرى من [[نظرية المجموعات]] بأنها أعلى وأقل قيم في المجموعة. يعد إيجاد النقاط العظمى والصغرى (الحرجة) نواة [[الإستمثال الرياضي]].
* الدالة x<sup>-x</sup> نهاية صغرى عامة عند ''x'' = 1/''e''.
* الدالة ''x''<sup>3</sup>/3 − ''x'' تكون مشتقتها الأولى ''x''<sup>2</sup> − 1 و مشتقتها الثانية 2''x''. بمساواة المشتق الأول بالصفر وحل المعادلة في ''x'' نحصل على قيم ساكنة عند at −1 و+1. من إشارة المشتق الثاني عند القيم نجد أن −1 عظمى محلية وأن +1 صغرى محلية. لاحظ أن هذه الدالة لاتملك نقاط عظمى أو صغرى عامة .
==Functions of more than one variable==<!-- This section is linked from [[Indifference curve]] -->
For functions of more than one variable, similar conditions apply. For example, in the (enlargeable) figure at the right, the necessary conditions for a ''local'' maximum are similar to those of a function with only one variable. The first [[partial derivatives]] as to ''z'' (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum because of the possibility of a [[saddle point]]. For use of these conditions to solve for a maximum, the function ''z'' must also be [[differentiable]] throughout. The [[second partial derivative test]] can help classify the point as a relative maximum or relative minimum.
In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if a differentiable function ''f'' defined on the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the [[intermediate value theorem]] and [[Rolle's theorem]] to prove this by [[reductio ad absurdum]]). In two and more dimensions, this argument fails, as the function
:<math>f(x,y)= x^2+y^2(1-x)^3,\qquad x,y\in\mathbb{R},</math>
shows. Its only critical point is at (0,0), which is a local minimum with ƒ(0,0) = 0. However, it cannot be a global one, because ƒ(4,1) = −11.
{|
|[[Image:MaximumParaboloid.png|thumb|300px|right|The global maximum is the point at the top]]
|[[Image:MaximumCounterexample.png|thumb|300px|left|Counterexample]]
|}
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== In relation to sets==
Maxima and minima are more generally defined for sets. In general, if an ordered set ''S'' has a greatest element ''m'', ''m'' is a maximal element. Furthermore, if ''S'' is a subset of an ordered set ''T'' and ''m'' is the greatest element of ''S'' with respect to order induced by ''T'', ''m'' is a least upper bound of ''S'' in ''T''. The similar result holds for least element, minimal element and greatest lower bound.
In the case of a general [[partial order]], the '''least element''' (smaller than all other) should not be confused with a '''minimal element''' (nothing is smaller). Likewise, a '''[[greatest element]]''' of a [[partially ordered set]] (poset) is an [[upper bound]] of the set which is contained within the set, whereas a '''maximal element''' ''m'' of a poset ''A'' is an element of ''A'' such that if ''m'' ≤ ''b'' (for any ''b'' in ''A'') then ''m'' = ''b''. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable.
In a [[total order|totally ordered]] set, or ''chain'', all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element and the maximal element will also be the greatest element. Thus in a totally ordered set we can simply use the terms '''''minimum''''' and '''''maximum'''''. If a chain is finite then it will always have a maximum and a minimum. If a chain is infinite then it need not have a maximum or a minimum. For example, the set of [[natural number]]s has no maximum, though it has a minimum. If an infinite chain ''S'' is bounded, then the [[topological closure|closure]] ''Cl(S)'' of the set occasionally has a minimum and a maximum, in such case they are called the '''[[infimum|greatest lower bound]]''' and the '''[[supremum|least upper bound]]''' of the set ''S'', respectively.
==انظر أيضا==
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