【正文】
知斷面面積的最大值一定存在,并且在區(qū)域:,內(nèi)取得,通過計算得知時的函數(shù)值比,時函數(shù)值為小,又函數(shù)在內(nèi)只有一個駐點,因此可以斷定,當(dāng),時,就能使斷面的面積最大。令 由于,上式為將代入(2)式得,再求出,則有,于是方程組的解是, 在考慮邊界,當(dāng)時,函數(shù)為的一元函數(shù),求最值點,由,得 。 解:函數(shù) 當(dāng)可見列表如下:(0,1)++單調(diào)性↑ ↓↑所以x=0為極大值點,極大值為0,x=1為極小點,極小值為3(如圖1)3-3圖(1) 極值的第二充分條件 求函數(shù)解:函數(shù)定義域為令x=6,如果當(dāng) 極值的第一充分條件和極值的第二充分條件 求函數(shù)解:,得又有。 (iii)時可能有極值,也可能沒有極值?! ?.第二充分條件設(shè)函數(shù)在點的某鄰域內(nèi)一階可導(dǎo),在點二階可導(dǎo),且,則 證明:由二階泰勒公式得=,所以 二元極值判別條件(1)必要條件設(shè)函數(shù)在點具有偏導(dǎo)數(shù),且在點處有極值,則它在該點處偏導(dǎo)數(shù)必然為零.有。 極值判別條件 一元極值判別條件(1)必要條件:費爾馬定理(2)充分條件 ?.第一充分條件設(shè)函數(shù)在點連續(xù),在鄰域和內(nèi)可導(dǎo),則 (i)在鄰域上,在鄰域上, (ii)在鄰域上,在鄰域上, 由導(dǎo)數(shù)的符號可知函數(shù)的單調(diào)性,故結(jié)論成立。 一元函數(shù)的極值定義:若函數(shù)在點可導(dǎo),則有費爾馬定理,點為極值點,而此時就是所謂的極值。而恰恰這些方法的終極解決,都?xì)w結(jié)于對函數(shù)極值和最值的求解。它為我們對于數(shù)學(xué)的進(jìn)一步研究起到很大幫助;同時,它對于其它相關(guān)學(xué)科的理解、學(xué)習(xí)與應(yīng)用也起著十分重要的作用,更對其他學(xué)科領(lǐng)域的展開有很大的促進(jìn)作用。等,:向量法,均值不等式法,換元法,消元法,柯西不等式法,數(shù)形結(jié)合法等, 關(guān)鍵詞:函數(shù),極值,最值,極值點,方法技巧. Abstract: in practical problems,often encounter a unary function. The function of two variables, and multiplefunctions of two yuan more than the most value questionand extremum problems and many other functions of mon problems. Extremum seeking a binary function,the main methods are: inequality extremum method,distribution method, derivation etc.. The value for theelement function, the main methods are: monotone method, function method, the discriminant method,plex method, derivative method, substitution methodetc.. For two yuan value function, the main methods are:conditional extremum of Lagrange multiplier method etc..Ask two yuan to the value function, the main methods are:mean inequality method, substitution method, partial derivative method etc.. For multivariate function, due to the increased number of variables,so that the more plicated the problem, find the function extreme value method mainly has: conditional extremum of multivariate Lagrange method, directional derivative, for multivariate function most value the most value problem with the function of one variable can be used to find the function extreme value is similar. The main methods are: vector method, the mean value inequality method, substitution, elimination method, the method of Cauchy inequality, the bination method,Keywords: function, extreme value, the value, extreme points, methods and techniques引言 作為函數(shù)性質(zhì)的一個重要分支和基本工具,函數(shù)極值和最值在數(shù)學(xué)與其他科學(xué)領(lǐng)域,如數(shù)學(xué)建模