极值(本地和绝对)据/h1>
已经有一个帐户?据一种href="//www.parkandroid.com/account/login/?next=/wiki/extrema/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link">这里登录。据/a>
测验据/h4>
相关...据/h4>
- 结石据/span>>据/span>
一个据strong>极值据/strong>一的(或极值)据一种href="//www.parkandroid.com/wiki/functions/" class="wiki_link" title="功能" target="_blank">功能据/a>是其在一些间隔中得到的函数的最大值或最小值的点。一种据strong>局部极值据/strong>函数的(或相对极值)是该功能的最大值或最小值的点据em>一些据/em>获得包含该点的开放间隔。据/p>
一个据strong>绝对极值据/strong>在一个函数的(或全局极值)据em>给予据/em>间隔是在其中获得最大或函数的最小值的点。通常情况下,时间间隔给出的函数的域,以及所述绝对极值是对应于整个函数的最大值或最小值的点。据/p>
极值(最大值和最小值)是很重要的,因为它们提供了大量的信息,关于应答功能和援助据一种href="//www.parkandroid.com/wiki/optimization-problems/" class="wiki_link" title="最优的问题" target="_blank">最优的问题据/a>。微积分提供了各种工具,以帮助快速确定极值的位置和性质。据/p>
全球极值据/h2>
一个点据span class="katex">
X据/span>是一个函数的绝对最大值或最小值据span class="katex">
F据/span>在间隔据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>如果据span class="katex">
F据/span>(据/span>X据/span>)据/span>≥据/span>F据/span>(据/span>X据/span>'据/span>)据/span>对所有人据span class="katex">
X据/span>'据/span>∈据/span>[据/span>一种据/span>那据/span>B.据/span>]据/span>或者如果据span class="katex">
F据/span>(据/span>X据/span>)据/span>≤.据/span>F据/span>(据/span>X据/span>'据/span>)据/span>对所有人据span class="katex">
X据/span>'据/span>∈据/span>[据/span>一种据/span>那据/span>B.据/span>]据/span>。点据span class="katex">
X据/span>是个据em>严格的据/em>(或唯一的)绝对最大值或最小值,如果它是满足这样的约束的唯一点。类似的定义保持区间据span class="katex">
[据/span>一种据/span>那据/span>∞据/span>)据/span>那据span class="katex">
(据/span>-据/span>∞据/span>那据/span>B.据/span>]据/span>, 和据span class="katex">
(据/span>-据/span>∞据/span>那据/span>∞据/span>)据/span>。间隔通常选择为域据span class="katex">
F据/span>。据/p>
如果该区域是在正或负方向上无限制或如果该函数不连续有可能不存在绝对的最大值或最小值。如果函数是不连续的(但为界),仍然会存在据一种href="//www.parkandroid.com/wiki/infimium/" class="wiki_link" title="确界或下确" target="_blank">确界或下确据/a>,但也有可能未必存在绝对的极值。如果函数是连续的和有界和间隔闭合,则一定存在的绝对最大和绝对最小值。据/p>
如果函数不连续,那么它可能在任何不连续点处具有绝对极值。通常,绝对极值对于最多有限数量的不连续性的功能仅适用。通过将这些点与以下方法一起考虑这些点,可以找到绝对极值。据/p>
如果函数是连续的,则绝对极值可以按照以下方法确定。给出一个功能据span class="katex">
F据/span>和间隔据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>那据/p>
对应于最大值的点据span class="katex">
F据/span>是绝对的最大值(最大值),以及对应于最小值的点据span class="katex">
F据/span>是绝对最小(极小)。其他值可能是局部极值。据/p>
确定在间隔的绝对最大值和下面的函数的最小值据span class="katex">
[据/span>-据/span>2据/span>3.据/span>那据/span>2据/span>7.据/span>]据/span>:据/span>
F据/span>(据/span>X据/span>)据/span>=据/span>⎩据/span>⎪据/span>⎪据/span>⎪据/span>⎨据/span>⎪据/span>⎪据/span>⎪据/span>⎧据/span>1据/span>-据/span>(据/span>X据/span>+据/span>1据/span>)据/span>2据/span>2据/span>X据/span>3.据/span>-据/span>(据/span>X据/span>-据/span>2据/span>)据/span>2据/span>3.据/span>-据/span>(据/span>X据/span>-据/span>2据/span>)据/span>3.据/span>X据/span>据据/span>0.据/span>0.据/span>≤.据/span>X据/span>≤.据/span>1据/span>1据/span>据据/span>X据/span>≤.据/span>2据/span>X据/span>>据/span>2据/span>。据/span>
该函数在临界点据span class="katex">
X据/span>=据/span>-据/span>1据/span>那据span class="katex">
X据/span>=据/span>0.据/span>那据span class="katex">
X据/span>=据/span>1据/span>, 和据span class="katex">
X据/span>=据/span>2据/span>。它在端点据span class="katex">
X据/span>=据/span>-据/span>2据/span>3.据/span>和据span class="katex">
X据/span>=据/span>2据/span>7.据/span>。据/p>
最大值的唯一可能性是据span class="katex">
X据/span>=据/span>-据/span>1据/span>那据span class="katex">
X据/span>=据/span>1据/span>, 和据span class="katex">
X据/span>=据/span>2据/span>。自从据span class="katex">
F据/span>(据/span>-据/span>1据/span>)据/span>=据/span>1据/span>那据span class="katex">
F据/span>(据/span>1据/span>)据/span>=据/span>2据/span>, 和据span class="katex">
F据/span>(据/span>2据/span>)据/span>=据/span>3.据/span>,绝对最大值位于据span class="katex">
(据/span>2据/span>那据/span>3.据/span>)据/span>。据/p>
最小值的唯一可能性是据span class="katex">
X据/span>=据/span>-据/span>2据/span>3.据/span>那据span class="katex">
X据/span>=据/span>0.据/span>那据span class="katex">
X据/span>=据/span>1据/span>, 和据span class="katex">
X据/span>=据/span>2据/span>7.据/span>。自从据span class="katex">
F据/span>(据/span>-据/span>2据/span>3.据/span>)据/span>=据/span>4.据/span>3.据/span>那据span class="katex">
F据/span>(据/span>0.据/span>)据/span>=据/span>0.据/span>那据span class="katex">
F据/span>(据/span>1据/span>)据/span>=据/span>2据/span>, 和据span class="katex">
F据/span>(据/span>2据/span>7.据/span>)据/span>=据/span>-据/span>8.据/span>3.据/span>,绝对最小值位于据span class="katex">
(据/span>2据/span>7.据/span>那据/span>-据/span>8.据/span>3.据/span>)据/span>。据span class="katex">
□据/span>
局部极值据/h2>
一个点据span class="katex">
X据/span>如果它是间隔中的函数的绝对最大值或最小值,则是函数的本地最大值或最小值据span class="katex">
(据/span>X据/span>-据/span>C据/span>那据/span>X据/span>+据/span>C据/span>)据/span>对于一些足够小的值据span class="katex">
C据/span>。据/p>
许多局部极值可识别功能的绝对最大值或最小值时被发现。据/p>
给出一个功能据span class="katex">
F据/span>和间隔据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>中,局部极值可以是不连续的点,非可微分的点或点是衍生物具有值据span class="katex">
0.据/span>。但是,这些点中都必须是局部极值的,因此必须为每个点检查该功能的本地行为。这是一个点据span class="katex">
X据/span>在间隔值的函数的据span class="katex">
(据/span>X据/span>-据/span>C据/span>那据/span>X据/span>+据/span>C据/span>)据/span>必须足够小的测试据span class="katex">
C据/span>。据/p>
如果函数是据一种href="//www.parkandroid.com/wiki/second-derivative-test/" class="wiki_link" title="两次微" target="_blank">两次微据/a>在据span class="katex">
X据/span>,再有就是提供一个较为简单的方法。据/p>
在间隔中对本地最大值和最小值进行分类据span class="katex">
[据/span>-据/span>2据/span>3.据/span>那据/span>2据/span>7.据/span>]据/span>:据/span>
F据/span>(据/span>X据/span>)据/span>=据/span>⎩据/span>⎪据/span>⎪据/span>⎪据/span>⎨据/span>⎪据/span>⎪据/span>⎪据/span>⎧据/span>1据/span>-据/span>(据/span>X据/span>+据/span>1据/span>)据/span>2据/span>2据/span>X据/span>3.据/span>-据/span>(据/span>X据/span>-据/span>2据/span>)据/span>2据/span>3.据/span>-据/span>(据/span>X据/span>-据/span>2据/span>)据/span>3.据/span>X据/span>据据/span>0.据/span>0.据/span>≤.据/span>X据/span>≤.据/span>1据/span>1据/span>据据/span>X据/span>≤.据/span>2据/span>X据/span>>据/span>2据/span>。据/span>
从图中看来,该功能前增加据span class="katex">
X据/span>=据/span>-据/span>1据/span>之间减少据span class="katex">
X据/span>=据/span>-据/span>1据/span>和据span class="katex">
X据/span>=据/span>0.据/span>,增加据span class="katex">
X据/span>=据/span>0.据/span>到据span class="katex">
X据/span>=据/span>2据/span>和减少后据span class="katex">
X据/span>=据/span>2据/span>。局部最大值位于据span class="katex">
X据/span>=据/span>-据/span>1据/span>和据span class="katex">
X据/span>=据/span>2据/span>。局部极小位于据span class="katex">
X据/span>=据/span>0.据/span>而在终点据span class="katex">
X据/span>=据/span>2据/span>7.据/span>。据/span>
□据/span>
什么是函数的所有局部极值的总和据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>|据/span>X据/span>|据/span>还是据/span> 遵守这一点据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>-据/span>X据/span>为了据span class="katex">
X据/span>据据/span>0.据/span>那据/span>
F据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>为了据span class="katex">
X据/span>=据/span>0.据/span>那据/span>和据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>为了据span class="katex">
X据/span>>据/span>0.据/span>。据/span>然后据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>-据/span>1据/span>据据/span>0.据/span>为了据span class="katex">
X据/span>据据/span>0.据/span>和据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>1据/span>>据/span>0.据/span>为了据span class="katex">
X据/span>>据/span>0.据/span>那据/span>这意味着该功能之前减少了据span class="katex">
X据/span>=据/span>0.据/span>并增加后据span class="katex">
X据/span>=据/span>0.据/span>。所以据span class="katex">
F据/span>(据/span>X据/span>)据/span>有一个地方的最低限度据span class="katex">
X据/span>=据/span>0.据/span>。据/span>由于当地最低值据span class="katex">
F据/span>(据/span>0.据/span>)据/span>=据/span>0.据/span>那据/span>所有本地极值的总和是据span class="katex">
0.据/span>。据/span>
□据/span>
实数的可能值是什么据span class="katex">
K.据/span>使得功能据/p>
F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>3.据/span>-据/span>2据/span>K.据/span>X据/span>2据/span>-据/span>4.据/span>K.据/span>X据/span>-据/span>1据/span>1据/span> 有没有极值?据/p>
区分据span class="katex">
F据/span>(据/span>X据/span>)据/span>关于据span class="katex">
X据/span>给据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>3.据/span>X据/span>2据/span>-据/span>4.据/span>K.据/span>X据/span>-据/span>4.据/span>K.据/span>。据/span>对于这个功能据span class="katex">
F据/span>(据/span>X据/span>)据/span>有没有极值,它必须是真实的,方程据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>有一个重复的根或非真实的复杂根。这相当于说等式的判别据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>3.据/span>X据/span>2据/span>-据/span>4.据/span>K.据/span>X据/span>-据/span>4.据/span>K.据/span>=据/span>0.据/span>必须是非正面的:据/p>
4.据/span>D.据/span>=据/span>(据/span>-据/span>2据/span>K.据/span>)据/span>2据/span>-据/span>3.据/span>⋅据/span>(据/span>-据/span>4.据/span>K.据/span>)据/span>=据/span>4.据/span>K.据/span>(据/span>K.据/span>+据/span>3.据/span>)据/span>≤.据/span>0.据/span>。据/span> 因此,范围据span class="katex">
K.据/span>这样据span class="katex">
F据/span>(据/span>X据/span>)据/span>没有极值据/p>
-据/span>3.据/span>≤.据/span>K.据/span>≤.据/span>0.据/span>。据/span>□据/span>
右边的图显示功能据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>|据/span>COS据/span>X据/span>+据/span>0.据/span>。据/span>5.据/span>|据/span>在间隔据span class="katex">
0.据/span>≤.据/span>X据/span>≤.据/span>1据/span>0.据/span>。据/p>
有多少本地极值是这个功能据span class="katex">
F据/span>(据/span>X据/span>)据/span>如果它的域名限制为据span class="katex">
0.据/span>≤.据/span>X据/span>≤.据/span>1据/span>0.据/span>还是据/span>
前 (据span class="katex">
一种据/span>据据/span>X据/span>)据/td>
后 (据span class="katex">
一种据/span>>据/span>X据/span>)据/td>
极值?据/td>
F据/span>(据/span>一种据/span>)据/span>据据/span>F据/span>(据/span>X据/span>)据/span>
F据/span>(据/span>一种据/span>)据/span>>据/span>F据/span>(据/span>X据/span>)据/span>
不据/td>
F据/span>(据/span>一种据/span>)据/span>据据/span>F据/span>(据/span>X据/span>)据/span>
F据/span>(据/span>一种据/span>)据/span>据据/span>F据/span>(据/span>X据/span>)据/span>
最大值据/td>
F据/span>(据/span>一种据/span>)据/span>>据/span>F据/span>(据/span>X据/span>)据/span>
F据/span>(据/span>一种据/span>)据/span>据据/span>F据/span>(据/span>X据/span>)据/span>
不据/td>
F据/span>(据/span>一种据/span>)据/span>>据/span>F据/span>(据/span>X据/span>)据/span>
F据/span>(据/span>一种据/span>)据/span>>据/span>F据/span>(据/span>X据/span>)据/span>
最低限度据/td>
F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>
极值?据/td>
积极的据/td>
最低限度据/td>
消极的据/td>
最大值据/td>
零据/td>
不据/td>
可怜的功能据/h2>
假设有问题的函数在间隔中是连续和微差的。然后,有一些快捷方式来确定极值。所有本地极值是衍生物为零的点(尽管衍生物可能为零,但对于不成为本地极值的点)。虽然它们仍然可以是端点(取决于所讨论的间隔),但是绝对极值也可以用几个快捷方式确定。这些是衍生物测试。据/p>
第一个衍生测试据/strong> 更简单,如果函数在之前增加并且在它之后减少,则一个点是一个功能。相反,如果函数,则一个点是最小的据em>减少据/em>之前并在它之后增加。据/p>
第二衍生物测试据/strong> 更简单,如果函数凹入函数,则点是一个函数的最大值,如果函数凹入,则一个点是函数的最小值。据/p>
衍生试验也可以施加到局部极值,给定足够小的间隔。实际上,第二衍生测试本身足以确定潜在的局部极值(对于可分辨率函数)是最大,最小的,或两者。据/p>
什么是所有的总和函数的局部极值据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>2据/span>X据/span>3.据/span>-据/span>6.据/span>X据/span>-据/span>3.据/span>还是据/span> 区分据span class="katex">
F据/span>(据/span>X据/span>)据/span>关于据span class="katex">
X据/span>给据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>6.据/span>X据/span>2据/span>-据/span>6.据/span>=据/span>6.据/span>(据/span>X据/span>+据/span>1据/span>)据/span>(据/span>X据/span>-据/span>1据/span>)据/span>。据/span>让据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>那据/span>然后据span class="katex">
X据/span>=据/span>-据/span>1据/span>那据/span>或者据span class="katex">
X据/span>=据/span>1据/span>。据/span>然后检查的符号据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>大约据span class="katex">
X据/span>=据/span>-据/span>1据/span>和据span class="katex">
X据/span>=据/span>1据/span>告诉我们据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>>据/span>0.据/span>为了据span class="katex">
X据/span>据据/span>-据/span>1据/span>那据/span>
F据/span>'据/span>(据/span>X据/span>)据/span>据据/span>0.据/span>为了据span class="katex">
-据/span>1据/span>据据/span>X据/span>据据/span>1据/span>那据/span>和据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>>据/span>0.据/span>为了据span class="katex">
X据/span>>据/span>1据/span>。据/span>这意味着据span class="katex">
F据/span>(据/span>X据/span>)据/span>有一个地方最大据span class="katex">
X据/span>=据/span>-据/span>1据/span>和地方最低限度据span class="katex">
X据/span>=据/span>1据/span>。据/span> 局部最大值的值是据span class="katex">
F据/span>(据/span>-据/span>1据/span>)据/span>=据/span>2据/span>⋅据/span>(据/span>-据/span>1据/span>)据/span>3.据/span>-据/span>6.据/span>⋅据/span>(据/span>-据/span>1据/span>)据/span>-据/span>3.据/span>=据/span>1据/span>。据/span>局部最小值的值是据span class="katex">
F据/span>(据/span>1据/span>)据/span>=据/span>2据/span>⋅据/span>(据/span>1据/span>)据/span>3.据/span>-据/span>6.据/span>⋅据/span>(据/span>1据/span>)据/span>-据/span>3.据/span>=据/span>-据/span>7.据/span>。据/span>因此,所有本地极值的总和是据span class="katex">
1据/span>-据/span>7.据/span>=据/span>-据/span>6.据/span>。据/span>
□据/span> 有多少本地极值是这个功能据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>(据/span>X据/span>-据/span>1据/span>)据/span>3.据/span>+据/span>5.据/span>有?据/p>
区分据span class="katex">
F据/span>(据/span>X据/span>)据/span>关于据span class="katex">
X据/span>给据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>3.据/span>(据/span>X据/span>-据/span>1据/span>)据/span>2据/span>。据/span>让据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>那据/span>然后据span class="katex">
X据/span>=据/span>1据/span>。据/span>然后检查的符号据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>大约据span class="katex">
X据/span>=据/span>1据/span>告诉我们据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>>据/span>0.据/span>为了据span class="katex">
X据/span>据据/span>1据/span>和据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>>据/span>0.据/span>为了据span class="katex">
X据/span>>据/span>1据/span>。据/span>这意味着据span class="katex">
F据/span>(据/span>X据/span>)据/span>与功能从来没有切换迹象的斜率没有局部极值。据/p>
因此,局部极值的数量是0。据span class="katex">
□据/span>
认为据span class="katex">
F据/span>是实值函数和据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>是在其上的间隔据span class="katex">
F据/span>定义和可微分。然后,如果据span class="katex">
C据/span>是一个关键点据span class="katex">
F据/span>在据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>那据/p>
认为据span class="katex">
F据/span>是实值函数和据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>是在其上的间隔据span class="katex">
F据/span>定义和两次可分辨。然后,如果据span class="katex">
C据/span>是一个关键点据span class="katex">
F据/span>在据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>那据/p>