极值点偏移问题(1)(2)(3)(4)(5)(6)(7)
极值点偏移问题一
——对称化构造(解题方法)
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三张图教你直观认识极值点偏移:
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例题展示
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点评:该题的三问由易到难,层层递进,完整展现了处理极值点偏移问题的一般方法——对称化构造的全过程,直观展示如下:
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把握以上三个关键点,就可以轻松解决一些极值点偏移问题.
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拓展
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小结:用对称化构造的方法解决极值点偏移问题大致分为以下三步:
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牛刀小试
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极值点偏移问题二
——函数的选取(操作细节)
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例题展示
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点评
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点评
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注1
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注2
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思考:上一讲极值点偏移问题(1)中练习1应该用哪一个函数来做呢?
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极值点偏移问题三
——变更结论(操作细节)
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例题展示
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解法一(换元法)
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解法二(加强命题)
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剧透:下一讲中我们还会给出这道题的第三种证法.
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能否将双变量的条件不等式化为单变量的函数不等式呢?
答案是肯定的,以笔者的学习经验为线索,我们先看一个例子.
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引例
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证明
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发现
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能否一开始就做这个代换呢?
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这样一种比值代换在极值点偏移问题中也大有可为.
下面就用这种方法再解前面举过的例子.
再解例1(3):
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再解例3:
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再解练习1:
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再解例4:
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再解例5:
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再解例7:
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再解例8:
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行文至此,相信读者已经领略到比值代换的威力.用比值代换解极值点偏移问题方便、快捷,简单得很.只需通过一个代换就可“双元”化“单元”,变为单变量的函数不等式,可证.那是不是可以就此忘掉前面三讲的内容呢?只需比值代换,就可偏移无忧?
这里,笔者必须指出,前面再解的过程中有意地略去了一些例子(不知细心的你是否发现),这就补上,请读者明察.
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试再解例2:
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试再解例6:
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试再解练习2:
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这是比值代换的败笔,又是最精彩之处.没有任何一种方法是万能的,我们不仅要熟悉它的优势,熟练它的操作,还要清醒地认识到它的缺陷,运用时要注意哪些问题,这其实是为了更好的运用.
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最后,我们来看比值代换另一个应用.
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牛刀小试
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极值点偏移问题五
——对数平均不等式(本质回归)
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回顾
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本讲要给的对数平均不等式是对基本不等式的加细.
对数平均不等式:
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先给出对数平均不等式的多种证法.
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证法1(对称化构造):
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证法2(比值代换):
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证法3(主元法):
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证法4(积分形式的柯西不等式):
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证法5(几何图示法):
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图1
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图2
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应用
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由对数平均不等式的证法1、2即可看出它与极值点偏移问题间千丝万缕的联系,下面就用对数平均不等式解前面举过的例题.
再解例1:
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再解例2:
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再解例3:
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再解练习1:
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再解例4:同本节例1
再解例5:同本节例1
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再解例7(2):
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再解例8:
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再解练习2:
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解练习3选项D:
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总 结
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极值点偏移问题,多与指数函数或对数函数有关,用对数平均不等式解题的关键有以下几步:
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细心的读者不难发现,用对数平均不等式来解极值点偏移问题的方法也有一定局限性,也不是万能的(再解过程中漏掉了例6,读者可尝试),其中能否简洁地表示出对数平均数是关键中的关键.
最后再举一例.
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证法1
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证法2
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极值点偏移问题六
——泰勒展开(本质回归)
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这一讲我们回到极值点偏移的直观图形上来,揭示极值点偏移问题的高等数学背景.以极小值点的偏移为例进行说明。
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图1
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图2
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以上只是直观(或者说非常粗略)的分析,下面拟用高等数学中的泰勒展开式进行严格证明,算作极值点偏移问题的另一种本质回归.
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极大值点的情形,推导过程同上,但结果却恰好相反,不再详述.
至此,我们得到极值点偏移问题的如下判定定理:
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注意:
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应用
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下面就用这个判定定理再解前面举过的例题.
再解例1:
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再解例2:
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再解例4:
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再解例6:
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再解例8:
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再解例10:
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——练习题及解答
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图1
练习题:
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提示与答案:
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来源:解忧高中数学杂货店