忘记密码?据/a>新用户?据一种href="//www.parkandroid.com/account/signup/?signup=true&next=/wiki/extreme-value-theorem/" id="problem-signup-link-alternative" class="btn-link ax-click" data-ax-id="clicked_signup_from_generic_modal" data-ax-type="button" data-next="/wiki/extreme-value-theorem/">注册据/a>
现有用户?据一种href="//www.parkandroid.com/account/login/?next=/wiki/extreme-value-theorem/" id="problem-login-link-alternative" class="btn-link ax-click" data-ax-id="clicked_login_from_generic_modal" data-ax-type="button" data-is_modal="true" data-next="/wiki/extreme-value-theorem/">登录据/a>
已经有一个帐户?据一种href="//www.parkandroid.com/account/login/?next=/wiki/extreme-value-theorem/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link">这里登录。据/a>
这据strong>极值定理据/strong>赋予存在据一种href="//www.parkandroid.com/wiki/extrema/" class="wiki_link" title="极值“target="_blank">极值据/a>A.据一种href="//www.parkandroid.com/wiki/continuous-functions/" class="wiki_link" title="连续功能“target="_blank">连续功能据/a>在关闭和有界间隔中定义。根据设置,可能需要决定存在,以及它们是否存在,并且给定函数的最大和最小(极端)值。例如,基于收集的数据的天气相关模型将分析用于解释风暴风速的手段。如果只测量高度和压力,则据一种href="//www.parkandroid.com/wiki/bernoullis-principle-fluids/" class="wiki_link" title="伯努利的原则“target="_blank">伯努利的原则据/a>,风行为是这些措施的结果。理解身高和最小压力将导致大约测量最大风速。在据一种href="//www.parkandroid.com/wiki/learn-and-practice-calculus-on-brilliant/" class="wiki_link" title="结石“target="_blank">结石据/a>,利用这一点据一种href="//www.parkandroid.com/wiki/derivative-by-first-principle/" class="wiki_link" title="衍生物“target="_blank">衍生物据/a>一个函数的(和缺乏它)为我们提供了一个找到这种极值的工具。据/p>
EVT的理论后果包括对微分功能研究的根本重要性的结果。最值得注意的是据一种href="//www.parkandroid.com/wiki/rolles-theorem/" class="wiki_link" title="罗勒的定理“target="_blank">罗勒的定理据/a>和据一种href="//www.parkandroid.com/wiki/mean-value-theorem/" class="wiki_link" title="平均值定理“target="_blank">平均值定理据/a>。据/p>
极值定理据/strong> 如果据一种href="//www.parkandroid.com/wiki/real-numbers/" class="wiki_link" title="实数“target="_blank">实数据/a> 一种据/mi> 一种据/annotation> 一种据/span>和据span class="katex"> B.据/mi> B.据/annotation> B.据/span>满足据span class="katex"> 一种据/mi> 据据/mo> B.据/mi> 那据/mo> a 一种据/span>据据/span>B.据/span>那据/span>和一个功能据span class="katex"> F据/mi> F据/annotation> F据/span>是持续的据span class="katex"> [据/mo> 一种据/mi> 那据/mo> B.据/mi> ]据/mo> 那据/mo> [A,B],据/annotation> [据/span>一种据/span>那据/span>B.据/span>]据/span>那据/span>然后据span class="katex"> F据/mi> F据/annotation> F据/span>达到其最大值和最小值据span class="katex"> [据/mo> 一种据/mi> 那据/mo> B.据/mi> ]据/mo> 。据/mi> [A,B]。据/annotation> [据/span>一种据/span>那据/span>B.据/span>]据/span>。据/span>
极值定理据/strong>
如果据一种href="//www.parkandroid.com/wiki/real-numbers/" class="wiki_link" title="实数“target="_blank">实数据/a> 一种据/mi> 一种据/annotation> 一种据/span>和据span class="katex"> B.据/mi> B.据/annotation> B.据/span>满足据span class="katex"> 一种据/mi> 据据/mo> B.据/mi> 那据/mo> a 一种据/span>据据/span>B.据/span>那据/span>和一个功能据span class="katex"> F据/mi> F据/annotation> F据/span>是持续的据span class="katex"> [据/mo> 一种据/mi> 那据/mo> B.据/mi> ]据/mo> 那据/mo> [A,B],据/annotation> [据/span>一种据/span>那据/span>B.据/span>]据/span>那据/span>然后据span class="katex"> F据/mi> F据/annotation> F据/span>达到其最大值和最小值据span class="katex"> [据/mo> 一种据/mi> 那据/mo> B.据/mi> ]据/mo> 。据/mi> [A,B]。据/annotation> [据/span>一种据/span>那据/span>B.据/span>]据/span>。据/span>
让据span class="katex"> F据/mi> F据/annotation> F据/span>在间隔内定义据span class="katex"> 一世据/mi> 。据/mi> 一世。据/annotation> 一世据/span>。据/span>
定义:据/p>
笔记:据/strong>该概念不能概括为没有限制的函数。如果我们放宽功能据span class="katex"> F据/mi> F据/annotation> F据/span>在无限的间隔上定义,如据span class="katex"> F据/mi> (据/mo> X据/mi> )据/mo> =据/mo> X据/mi> 2据/mn> f(x)= x ^ 2据/annotation> F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>2据/span>在据span class="katex"> 一世据/mi> =据/mo> [据/mo> 0.据/mn> 那据/mo> ∞据/mi> )据/mo> 我= [0,\ idty)据/annotation> 一世据/span>=据/span>[据/span>0.据/span>那据/span>∞据/span>)据/span>。然后这个函数最小据span class="katex"> X据/mi> =据/mo> 0.据/mn> x = 0.据/annotation> X据/span>=据/span>0.据/span>但由于它正在增加据span class="katex"> 一世据/mi> 一世据/annotation> 一世据/span>然后,它不会肆无忌惮地生长据一种href="//www.parkandroid.com/wiki/infinity/" class="wiki_link" title="无限“target="_blank">无限据/a>。如果我们放宽功能据span class="katex"> F据/mi> F据/annotation> F据/span>在非校准的间隔上定义,如据span class="katex"> F据/mi> (据/mo> X据/mi> )据/mo> =据/mo> 1据/mn> X据/mi> f(x)= \ frac1x据/annotation> F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>1据/span>在据span class="katex"> 一世据/mi> =据/mo> (据/mo> 0.据/mn> 那据/mo> 1据/mn> ]据/mo> 那据/mo> 我=(0,1],据/annotation> 一世据/span>=据/span>(据/span>0.据/span>那据/span>1据/span>]据/span>那据/span>然后这个函数最小据span class="katex"> X据/mi> =据/mo> 1据/mn> x = 1据/annotation> X据/span>=据/span>1据/span>但它的时候会变得不平衡据span class="katex"> X据/mi> X据/annotation> X据/span>方法据span class="katex"> 0.据/mn> 0.据/annotation> 0.据/span>从右边。最后,如果我们放宽功能据span class="katex"> F据/mi> F据/annotation> F据/span>有一个据一种target="_blank" rel="nofollow" href="//www.parkandroid.com/wiki/continuous-functions/">不连续据/a>(参见链接中的示例#2),就像据span class="katex"> F据/mi> (据/mo> X据/mi> )据/mo> =据/mo> X据/mi> 2据/mn> f(x)= x ^ 2据/annotation> F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>2据/span>在据span class="katex"> [据/mo> -据/mo> 1据/mn> 那据/mo> 0.据/mn> )据/mo> ∪据/mo> (据/mo> 0.据/mn> 那据/mo> 1据/mn> ]据/mo> [-1,0)\杯(0,1]据/annotation> [据/span>-据/span>1据/span>那据/span>0.据/span>)据/span>∪据/span>(据/span>0.据/span>那据/span>1据/span>]据/span>和据span class="katex"> F据/mi> (据/mo> 0.据/mn> )据/mo> =据/mo> 1据/mn> 2据/mn> 那据/mo> F(0)= \ FRAC12,据/annotation> F据/span>(据/span>0.据/span>)据/span>=据/span>2据/span>1据/span>那据/span>然后这个函数最大值据span class="katex"> X据/mi> =据/mo> -据/mo> 1据/mn> x = -1据/annotation> X据/span>=据/span>-据/span>1据/span>和据span class="katex"> X据/mi> =据/mo> 1据/mn> x = 1据/annotation> X据/span>=据/span>1据/span>但它没有任何影响据span class="katex"> C据/mi> C据/annotation> C据/span>在据span class="katex"> (据/mo> 0.据/mn> 那据/mo> 1据/mn> )据/mo> (0,1)据/annotation> (据/span>0.据/span>那据/span>1据/span>)据/span>有据span class="katex"> D.据/mi> D.据/annotation> D.据/span>在据span class="katex"> (据/mo> 0.据/mn> 那据/mo> C据/mi> )据/mo> (0,c)据/annotation> (据/span>0.据/span>那据/span>C据/span>)据/span>这样据span class="katex"> F据/mi> (据/mo> D.据/mi> )据/mo> 据据/mo> F据/mi> (据/mo> C据/mi> )据/mo> f(d) F据/span>(据/span>D.据/span>)据/span>据据/span>F据/span>(据/span>C据/span>)据/span>。据/p>
如引言中所提到的,知道功能的存在和计算极值将解决许多设置中的问题。存在于极值定理给出,因此焦点现在正在讨论如何计算这些值。据/p>
问题:“我们如何找到连续功能的最大值和最小值据span class="katex"> F据/mi> F据/annotation> F据/span>在关闭和有界间隔据span class="katex"> [据/mo> 一种据/mi> 那据/mo> B.据/mi> ]据/mo> 还是据/mo> [A,B]?据/annotation> [据/span>一种据/span>那据/span>B.据/span>]据/span>还是据/span>“据/p> 找到衍生物据span class="katex"> F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> f'(x)据/annotation> F据/span>'据/span>(据/span>X据/span>)据/span>的据span class="katex"> F据/mi> F据/annotation> F据/span>。据/li> 找到所有临界值据span class="katex"> F据/mi> F据/annotation> F据/span>,让我们说据span class="katex"> C据/mi> 1据/mn> 那据/mo> C据/mi> 2据/mn> 那据/mo> ......据/mo> 那据/mo> C据/mi> K.据/mi> 。据/mi> C_1,C_2,...,C_K。据/annotation> C据/span>1据/span>那据/span>C据/span>2据/span>那据/span>......据/span>那据/span>C据/span>K.据/span>。据/span> 计算据span class="katex"> F据/mi> (据/mo> 一种据/mi> )据/mo> F A)据/annotation> F据/span>(据/span>一种据/span>)据/span>和据span class="katex"> F据/mi> (据/mo> B.据/mi> )据/mo> F(b)据/annotation> F据/span>(据/span>B.据/span>)据/span>, 和...一起据span class="katex"> F据/mi> (据/mo> C据/mi> 1据/mn> )据/mo> 那据/mo> F据/mi> (据/mo> C据/mi> 2据/mn> )据/mo> 那据/mo> ......据/mo> 那据/mo> F据/mi> (据/mo> C据/mi> K.据/mi> )据/mo> 。据/mi> f(c_1),f(c_2),...,f(c_k)。据/annotation> F据/span>(据/span>C据/span>1据/span>)据/span>那据/span>F据/span>(据/span>C据/span>2据/span>)据/span>那据/span>......据/span>那据/span>F据/span>(据/span>C据/span>K.据/span>)据/span>。据/span> 比较(订购)步骤3中的值。据/li> 然后,最小值是步骤4中值的较小值,最大值是较大的。据/li>
鉴于功能据span class="katex"> F据/mi> (据/mo> X据/mi> )据/mo> =据/mo> X据/mi> 3.据/mn> -据/mo> 3.据/mn> X据/mi> 2据/mn> +据/mo> 3.据/mn> X据/mi> +据/mo> 1据/mn> f(x)= x ^ 3-3x ^ 2 + 3x + 1据/annotation> F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>3.据/span>-据/span>3.据/span>X据/span>2据/span>+据/span>3.据/span>X据/span>+据/span>1据/span>,在间隔内找到此功能的最大值和最小值据span class="katex"> [据/mo> 0.据/mn> 那据/mo> 2据/mn> ]据/mo> [0,2]据/annotation> [据/span>0.据/span>那据/span>2据/span>]据/span>。据/p> 此功能是可微分的,因此连续,问题间隔是关闭和界限的。因此,我们可以应用EVT。据/p> 我们采取了这个功能的第一个衍生物:据/p> F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> =据/mo> 3.据/mn> X据/mi> 2据/mn> -据/mo> 6.据/mn> X据/mi> +据/mo> 3.据/mn> =据/mo> 3.据/mn> (据/mo> X据/mi> 2据/mn> -据/mo> 2据/mn> X据/mi> +据/mo> 1据/mn> )据/mo> 。据/mi> \ begin {对齐} f'(x)&= 3x ^ 2-6x + 3 \\&= 3(x ^ 2-2x + 1)。\结束{对齐}据/annotation> 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>6.据/span>X据/span>+据/span>3.据/span>=据/span>3.据/span>(据/span>X据/span>2据/span>-据/span>2据/span>X据/span>+据/span>1据/span>)据/span>。据/span> 自从据span class="katex"> F据/mi> F据/annotation> F据/span>在其整个域上是可差的,我们只需要找到相对极值并评估间隔的终点。环境据span class="katex"> F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> =据/mo> 0.据/mn> f'(x)= 0据/annotation> F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>并解决,我们发现唯一的关键号码是据span class="katex"> X据/mi> 1据/mn> =据/mo> 1据/mn> 那据/mo> x_1 = 1,据/annotation> X据/span>1据/span>=据/span>1据/span>那据/span>这给出了价值据span class="katex"> y据/mi> 1据/mn> =据/mo> 2。据/mn> y_1 = 2。据/annotation> y据/span>1据/span>=据/span>2据/span>。据/span> 警告:据/strong>它是相对extrema只发生在关键号码时的微积分定理。问题是确定哪些关键数字给我们相对的极值。这是以两种方式完成的:据一种target="_blank" rel="nofollow" href="//www.parkandroid.com/wiki/extrema/">第一个衍生测试据/strong>或者据一种target="_blank" rel="nofollow" href="//www.parkandroid.com/wiki/extrema/">第二衍生物测试据/strong>。通过第一衍生物测试,据span class="katex"> F据/mi> '据/mo> F'据/annotation> F据/span>'据/span>在关键号之前和之后不会改变标志据span class="katex"> X据/mi> 1据/mn> =据/mo> 1据/mn> X_1 = 1据/annotation> X据/span>1据/span>=据/span>1据/span>。因此,没有与此数量相关的相对极值。据/p> 计算端点,我们有据span class="katex"> y据/mi> 2据/mn> =据/mo> F据/mi> (据/mo> 0.据/mn> )据/mo> =据/mo> 1据/mn> y_2 = f(0)= 1据/annotation> y据/span>2据/span>=据/span>F据/span>(据/span>0.据/span>)据/span>=据/span>1据/span>和据span class="katex"> y据/mi> 3.据/mn> =据/mo> F据/mi> (据/mo> 2据/mn> )据/mo> =据/mo> 3.据/mn> Y_3 = F(2)= 3据/annotation> y据/span>3.据/span>=据/span>F据/span>(据/span>2据/span>)据/span>=据/span>3.据/span>。因此,最小值据span class="katex"> F据/mi> F据/annotation> F据/span>在据span class="katex"> [据/mo> 0.据/mn> 那据/mo> 2据/mn> ]据/mo> [0,2]据/annotation> [据/span>0.据/span>那据/span>2据/span>]据/span>是1,其最大间隔为3。据span class="katex"> □据/mi> _\正方形据/annotation> □据/span>
此功能是可微分的,因此连续,问题间隔是关闭和界限的。因此,我们可以应用EVT。据/p>
我们采取了这个功能的第一个衍生物:据/p>
F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> =据/mo> 3.据/mn> X据/mi> 2据/mn> -据/mo> 6.据/mn> X据/mi> +据/mo> 3.据/mn> =据/mo> 3.据/mn> (据/mo> X据/mi> 2据/mn> -据/mo> 2据/mn> X据/mi> +据/mo> 1据/mn> )据/mo> 。据/mi> \ begin {对齐} f'(x)&= 3x ^ 2-6x + 3 \\&= 3(x ^ 2-2x + 1)。\结束{对齐}据/annotation> 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>6.据/span>X据/span>+据/span>3.据/span>=据/span>3.据/span>(据/span>X据/span>2据/span>-据/span>2据/span>X据/span>+据/span>1据/span>)据/span>。据/span>
自从据span class="katex"> F据/mi> F据/annotation> F据/span>在其整个域上是可差的,我们只需要找到相对极值并评估间隔的终点。环境据span class="katex"> F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> =据/mo> 0.据/mn> f'(x)= 0据/annotation> F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>并解决,我们发现唯一的关键号码是据span class="katex"> X据/mi> 1据/mn> =据/mo> 1据/mn> 那据/mo> x_1 = 1,据/annotation> X据/span>1据/span>=据/span>1据/span>那据/span>这给出了价值据span class="katex"> y据/mi> 1据/mn> =据/mo> 2。据/mn> y_1 = 2。据/annotation> y据/span>1据/span>=据/span>2据/span>。据/span>
警告:据/strong>它是相对extrema只发生在关键号码时的微积分定理。问题是确定哪些关键数字给我们相对的极值。这是以两种方式完成的:据一种target="_blank" rel="nofollow" href="//www.parkandroid.com/wiki/extrema/">第一个衍生测试据/strong>或者据一种target="_blank" rel="nofollow" href="//www.parkandroid.com/wiki/extrema/">第二衍生物测试据/strong>。通过第一衍生物测试,据span class="katex"> F据/mi> '据/mo> F'据/annotation> F据/span>'据/span>在关键号之前和之后不会改变标志据span class="katex"> X据/mi> 1据/mn> =据/mo> 1据/mn> X_1 = 1据/annotation> X据/span>1据/span>=据/span>1据/span>。因此,没有与此数量相关的相对极值。据/p> 计算端点,我们有据span class="katex"> y据/mi> 2据/mn> =据/mo> F据/mi> (据/mo> 0.据/mn> )据/mo> =据/mo> 1据/mn> y_2 = f(0)= 1据/annotation> y据/span>2据/span>=据/span>F据/span>(据/span>0.据/span>)据/span>=据/span>1据/span>和据span class="katex"> y据/mi> 3.据/mn> =据/mo> F据/mi> (据/mo> 2据/mn> )据/mo> =据/mo> 3.据/mn> Y_3 = F(2)= 3据/annotation> y据/span>3.据/span>=据/span>F据/span>(据/span>2据/span>)据/span>=据/span>3.据/span>。因此,最小值据span class="katex"> F据/mi> F据/annotation> F据/span>在据span class="katex"> [据/mo> 0.据/mn> 那据/mo> 2据/mn> ]据/mo> [0,2]据/annotation> [据/span>0.据/span>那据/span>2据/span>]据/span>是1,其最大间隔为3。据span class="katex"> □据/mi> _\正方形据/annotation> □据/span>
计算端点,我们有据span class="katex"> y据/mi> 2据/mn> =据/mo> F据/mi> (据/mo> 0.据/mn> )据/mo> =据/mo> 1据/mn> y_2 = f(0)= 1据/annotation> y据/span>2据/span>=据/span>F据/span>(据/span>0.据/span>)据/span>=据/span>1据/span>和据span class="katex"> y据/mi> 3.据/mn> =据/mo> F据/mi> (据/mo> 2据/mn> )据/mo> =据/mo> 3.据/mn> Y_3 = F(2)= 3据/annotation> y据/span>3.据/span>=据/span>F据/span>(据/span>2据/span>)据/span>=据/span>3.据/span>。因此,最小值据span class="katex"> F据/mi> F据/annotation> F据/span>在据span class="katex"> [据/mo> 0.据/mn> 那据/mo> 2据/mn> ]据/mo> [0,2]据/annotation> [据/span>0.据/span>那据/span>2据/span>]据/span>是1,其最大间隔为3。据span class="katex"> □据/mi> _\正方形据/annotation> □据/span>
鉴于功能据span class="katex"> F据/mi> (据/mo> X据/mi> )据/mo> =据/mo> X据/mi> 3.据/mn> -据/mo> 9.据/mn> 2据/mn> X据/mi> 2据/mn> -据/mo> 12.据/mn> X据/mi> +据/mo> 20.据/mn> f(x)= x ^ 3- \ frac {9} {2} x ^ 2-12x + 20据/annotation> F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>3.据/span>-据/span>2据/span>9.据/span>X据/span>2据/span>-据/span>1据/span>2据/span>X据/span>+据/span>2据/span>0.据/span>,在间隔内找到此功能的最大值和最小值据span class="katex"> [据/mo> -据/mo> 2据/mn> 那据/mo> 6.据/mn> ]据/mo> [-2,6]据/annotation> [据/span>-据/span>2据/span>那据/span>6.据/span>]据/span>。据/p> 此功能是可微分的,因此连续,问题间隔是关闭和界限的。因此,我们可以将EVT应用于此设置。据/p> 我们采取了这个功能的第一个衍生物:据/p> F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> =据/mo> 3.据/mn> X据/mi> 2据/mn> -据/mo> 9.据/mn> X据/mi> -据/mo> 12.据/mn> =据/mo> 3.据/mn> (据/mo> X据/mi> 2据/mn> -据/mo> 3.据/mn> X据/mi> -据/mo> 4.据/mn> )据/mo> 。据/mi> \ begin {对齐} f'(x)&= 3x ^ 2-9x-12 \\&= 3(x ^ 2-3x-4)。\结束{对齐}据/annotation> 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>9.据/span>X据/span>-据/span>1据/span>2据/span>=据/span>3.据/span>(据/span>X据/span>2据/span>-据/span>3.据/span>X据/span>-据/span>4.据/span>)据/span>。据/span> 自从据span class="katex"> F据/mi> F据/annotation> F据/span>在其整个域上是可差的,我们只需要找到相对极值并评估间隔的终点。环境据span class="katex"> F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> =据/mo> 0.据/mn> f'(x)= 0据/annotation> F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>并解决,我们发现关键数字是据span class="katex"> X据/mi> 1据/mn> =据/mo> -据/mo> 1据/mn> X_1 = -1据/annotation> X据/span>1据/span>=据/span>-据/span>1据/span>和据span class="katex"> X据/mi> 2据/mn> =据/mo> 4.据/mn> 那据/mo> x_2 = 4,据/annotation> X据/span>2据/span>=据/span>4.据/span>那据/span>由第一衍生物测试提供相对极值的值据span class="katex"> y据/mi> 1据/mn> =据/mo> 26.5据/mn> Y_1 = 26.5.据/annotation> y据/span>1据/span>=据/span>2据/span>6.据/span>。据/span>5.据/span>和据span class="katex"> y据/mi> 2据/mn> =据/mo> -据/mo> 36.据/mn> Y_2 = -36.据/annotation> y据/span>2据/span>=据/span>-据/span>3.据/span>6.据/span>。计算端点,我们有据span class="katex"> y据/mi> 3.据/mn> =据/mo> F据/mi> (据/mo> -据/mo> 2据/mn> )据/mo> =据/mo> 18.据/mn> Y_3 = F(-2)= 18据/annotation> y据/span>3.据/span>=据/span>F据/span>(据/span>-据/span>2据/span>)据/span>=据/span>1据/span>8.据/span>和据span class="katex"> y据/mi> 4.据/mn> =据/mo> F据/mi> (据/mo> 6.据/mn> )据/mo> =据/mo> 2据/mn> Y_4 = F(6)= 2据/annotation> y据/span>4.据/span>=据/span>F据/span>(据/span>6.据/span>)据/span>=据/span>2据/span>。因此,最小值据span class="katex"> F据/mi> F据/annotation> F据/span>在据span class="katex"> [据/mo> -据/mo> 2据/mn> 那据/mo> 6.据/mn> ]据/mo> [-2,6]据/annotation> [据/span>-据/span>2据/span>那据/span>6.据/span>]据/span>是-36,其最大间隔为26.5。据span class="katex"> □据/mi> _\正方形据/annotation> □据/span>
此功能是可微分的,因此连续,问题间隔是关闭和界限的。因此,我们可以将EVT应用于此设置。据/p>
F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> =据/mo> 3.据/mn> X据/mi> 2据/mn> -据/mo> 9.据/mn> X据/mi> -据/mo> 12.据/mn> =据/mo> 3.据/mn> (据/mo> X据/mi> 2据/mn> -据/mo> 3.据/mn> X据/mi> -据/mo> 4.据/mn> )据/mo> 。据/mi> \ begin {对齐} f'(x)&= 3x ^ 2-9x-12 \\&= 3(x ^ 2-3x-4)。\结束{对齐}据/annotation> 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>9.据/span>X据/span>-据/span>1据/span>2据/span>=据/span>3.据/span>(据/span>X据/span>2据/span>-据/span>3.据/span>X据/span>-据/span>4.据/span>)据/span>。据/span>
自从据span class="katex"> F据/mi> F据/annotation> F据/span>在其整个域上是可差的,我们只需要找到相对极值并评估间隔的终点。环境据span class="katex"> F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> =据/mo> 0.据/mn> f'(x)= 0据/annotation> F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>并解决,我们发现关键数字是据span class="katex"> X据/mi> 1据/mn> =据/mo> -据/mo> 1据/mn> X_1 = -1据/annotation> X据/span>1据/span>=据/span>-据/span>1据/span>和据span class="katex"> X据/mi> 2据/mn> =据/mo> 4.据/mn> 那据/mo> x_2 = 4,据/annotation> X据/span>2据/span>=据/span>4.据/span>那据/span>由第一衍生物测试提供相对极值的值据span class="katex"> y据/mi> 1据/mn> =据/mo> 26.5据/mn> Y_1 = 26.5.据/annotation> y据/span>1据/span>=据/span>2据/span>6.据/span>。据/span>5.据/span>和据span class="katex"> y据/mi> 2据/mn> =据/mo> -据/mo> 36.据/mn> Y_2 = -36.据/annotation> y据/span>2据/span>=据/span>-据/span>3.据/span>6.据/span>。计算端点,我们有据span class="katex"> y据/mi> 3.据/mn> =据/mo> F据/mi> (据/mo> -据/mo> 2据/mn> )据/mo> =据/mo> 18.据/mn> Y_3 = F(-2)= 18据/annotation> y据/span>3.据/span>=据/span>F据/span>(据/span>-据/span>2据/span>)据/span>=据/span>1据/span>8.据/span>和据span class="katex"> y据/mi> 4.据/mn> =据/mo> F据/mi> (据/mo> 6.据/mn> )据/mo> =据/mo> 2据/mn> Y_4 = F(6)= 2据/annotation> y据/span>4.据/span>=据/span>F据/span>(据/span>6.据/span>)据/span>=据/span>2据/span>。因此,最小值据span class="katex"> F据/mi> F据/annotation> F据/span>在据span class="katex"> [据/mo> -据/mo> 2据/mn> 那据/mo> 6.据/mn> ]据/mo> [-2,6]据/annotation> [据/span>-据/span>2据/span>那据/span>6.据/span>]据/span>是-36,其最大间隔为26.5。据span class="katex"> □据/mi> _\正方形据/annotation> □据/span>
鉴于功能据span class="katex"> F据/mi> (据/mo> X据/mi> )据/mo> =据/mo> 3.据/mn> X据/mi> X据/mi> 2据/mn> -据/mo> 1据/mn> f(x)= \ frac {3x} {x ^ 2-1}据/annotation> F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>2据/span>-据/span>1据/span>3.据/span>X据/span>,找到关键的数量据span class="katex"> F据/mi> F据/annotation> F据/span>。据/p> 我们采取了这个功能的第一个衍生物:据/p> F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> =据/mo> 3.据/mn> (据/mo> X据/mi> 2据/mn> +据/mo> 1据/mn> )据/mo> (据/mo> 1据/mn> -据/mo> X据/mi> 2据/mn> )据/mo> 2据/mn> 。据/mi> \ begin {对齐} f'(x)= \ frac {3(x ^ 2 + 1)} {({1-x ^ 2})^ 2}。\结束{对齐}据/annotation> F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>(据/span>1据/span>-据/span>X据/span>2据/span>)据/span>2据/span>3.据/span>(据/span>X据/span>2据/span>+据/span>1据/span>)据/span>。据/span> 遵守这一点据span class="katex"> F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> =据/mo> 0.据/mn> f'(x)= 0据/annotation> F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>没有解决真实数字的解决方案据span class="katex"> F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> f'(x)据/annotation> F据/span>'据/span>(据/span>X据/span>)据/span>是未定义的据span class="katex"> X据/mi> =据/mo> -据/mo> 1据/mn> x = -1据/annotation> X据/span>=据/span>-据/span>1据/span>和据span class="katex"> X据/mi> =据/mo> 1据/mn> x = 1据/annotation> X据/span>=据/span>1据/span>。但请注意数字据span class="katex"> X据/mi> =据/mo> -据/mo> 1据/mn> x = -1据/annotation> X据/span>=据/span>-据/span>1据/span>和据span class="katex"> X据/mi> =据/mo> 1据/mn> x = 1据/annotation> X据/span>=据/span>1据/span>不是关键号码,因为它们不在域名据span class="katex"> F据/mi> F据/annotation> F据/span>。据/p> 因此,此功能没有关键数字。据span class="katex"> □据/mi> _\正方形据/annotation> □据/span>
F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> =据/mo> 3.据/mn> (据/mo> X据/mi> 2据/mn> +据/mo> 1据/mn> )据/mo> (据/mo> 1据/mn> -据/mo> X据/mi> 2据/mn> )据/mo> 2据/mn> 。据/mi> \ begin {对齐} f'(x)= \ frac {3(x ^ 2 + 1)} {({1-x ^ 2})^ 2}。\结束{对齐}据/annotation> F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>(据/span>1据/span>-据/span>X据/span>2据/span>)据/span>2据/span>3.据/span>(据/span>X据/span>2据/span>+据/span>1据/span>)据/span>。据/span>
遵守这一点据span class="katex"> F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> =据/mo> 0.据/mn> f'(x)= 0据/annotation> F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>没有解决真实数字的解决方案据span class="katex"> F据/mi> '据/mo> (据/mo> X据/mi> )据/mo> f'(x)据/annotation> F据/span>'据/span>(据/span>X据/span>)据/span>是未定义的据span class="katex"> X据/mi> =据/mo> -据/mo> 1据/mn> x = -1据/annotation> X据/span>=据/span>-据/span>1据/span>和据span class="katex"> X据/mi> =据/mo> 1据/mn> x = 1据/annotation> X据/span>=据/span>1据/span>。但请注意数字据span class="katex"> X据/mi> =据/mo> -据/mo> 1据/mn> x = -1据/annotation> X据/span>=据/span>-据/span>1据/span>和据span class="katex"> X据/mi> =据/mo> 1据/mn> x = 1据/annotation> X据/span>=据/span>1据/span>不是关键号码,因为它们不在域名据span class="katex"> F据/mi> F据/annotation> F据/span>。据/p> 因此,此功能没有关键数字。据span class="katex"> □据/mi> _\正方形据/annotation> □据/span>
因此,此功能没有关键数字。据span class="katex"> □据/mi> _\正方形据/annotation> □据/span>
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