最大和最低

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A global maximum refers to the point with the largest  $\begin{array}{r}y\end{array}$ value possible on a function. A global minimum refers to the point with the smallest  $\begin{array}{r}y\end{array}$ value possible. Together these two values are referred to as global extrema . There can only be one global maximum and only one global minimum. Global refers to entire space where the function is defined. Global extrema are also called absolute extrema .
::全球最大值指函数上最大值为y值的点。全球最低值指最小值为y值的点。这两个值加在一起被称为全球极限值。只能有一个全球最大值,而全球最小值只有一个。全球最大值指函数定义的整个空间。全球极限也称为绝对极限值。

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In addition to global maximums and global minimums, there are also local extrema or relative maximums and relative minimums . The word relative is used because in relation to some neighborhood, t hese values stand out as being the highest or the lowest. 
::除了全球最大值和全球最低值之外,还有局部极端值或相对最高值和相对最低值。 使用相对值是因为相对于某些邻里而言,这些值是最高的或最低的。

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Calculus uses advanced analytic tools to compute extreme values, but for the purposes of PreCalculus it is sufficient to be able to identify and categorize extreme values graphically or through the use of technology. For example, the TI-84 has a maximum finder when you select <2 ^nd > then .
::微积分使用先进的分析工具来计算极端值,但为了预微积分的目的,只需能够以图形方式或通过使用技术来识别和分类极端值即可。例如,当您选择 < 2nd> 时, TI- 84 具有一个最大查找器。

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Examples
::实例

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Example 1
::例1

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Earlier, you were asked to identify and distinguish between different peaks and valleys on a graph. Maximums and minimums should be intuitive because they simply identify the highest points and the lowest points, or the peaks and the valleys, in a graph. There is a formal distinction about whether a maximum is the highest on some local open interval (does not matter how small), or whether it is simply the highest overall. 
::早些时候,您被要求在图表中辨别和区分不同的峰值和谷值。 最大值和最小值应该是直观的, 因为它们仅仅是在图表中标出最高点和最低点, 或峰值和谷值。 在某个局部开放间隔中,最高值是最高点( 不论大小) , 还是最高点是最高点方面, 存在正式的区别 。

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Identify and categorize all extrema. 
::识别并分类所有extrema。

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There are no global or local maximums or minimums. The function flattens, but does not actually reach a peak or a valley. 
::没有全球或本地的上限或最小值。 函数平坦, 但实际上没有达到峰值或山谷 。

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Example 3
::例3

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Identify and categorize all the extrema in the graph below:
::在下图中标明并分类所有extrema:

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Since the function appears from the arrows to increase and decrease beyond the display, there are no global extrema. There is a local maximum at approximately (0, 3) and a local minimum at approximately (2.8, -7). 
::由于该函数从箭头中显示到显示外的增减,因此没有全球外缘。本地最大值大约为(0,3),当地最低值大约为(2.8,7)。

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Example 4
::例4

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Identify and categorize all the extrema in the graph below:
::在下图中标明并分类所有extrema:

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Since the function seems to abruptly end at the end points and does not go beyond the display, the endpoints are important.
::由于该函数似乎在结束点突然结束,而且没有超出显示范围,所以端点是重要的。

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There is a global minimum at (0, 0). There is a local maximum at (-1, 1) and a global maximum at (5, 5). 
::全球最低值为(0,0),当地最高值为(1,1),全球最高值为(5,5)。

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Example 5
::例5

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Identify and categorize all the extrema in the graph below: 
::在下图中标明并分类所有extrema:

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Since this function appears to increase to the right as indicated by the arrow there is no global maximum. There are not any other high points either, so there are no local maximums. There is only the end point at (0, 0) which is a global minimum.
::由于此函数似乎如箭头所示向右增加, 没有全球最大值 。 也没有其它高点, 所以没有本地最大值 。 只有端点( 0, 0) 是全球最低值 。

Summary

播放段落 Global maximum and global minimum refer to the largest and smallest y values possible on a function, respectively, and are collectively called global extrema or absolute extrema. ::全球最高和全球最低数值分别指一个函数上可能的最大值和最小值y,统称为全球极限值或绝对极限值。 播放段落 Local extrema  also known as relative maximums and relative minimums, are values that stand out as the highest or lowest in a specific neighborhood of the function. ::局部极限也称为相对上限和相对最小值,其值在函数的特定区域中是最高或最低的。

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Use the graph below for 1-2.
::1-2使用下图。

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1. Identify any global extrema.
::1. 查明任何全球极端。

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2. Identify any local extrema.
::2. 标明任何局部端口。

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Use the graph below for 3-4.
::3-4使用下图。

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3. Identify any global extrema.
::3. 查明任何全球极端。

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4. Identify any local extrema.
::4. 标明任何局部端口。

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Use the graph below for 5-6.
::5-6使用下图。

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5. Identify any global extrema.
::5. 查明任何全球极端。

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6. Identify any local extrema.
::6. 标明任何局部端口。

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Use the graph below for 7-8.
::7-8使用下图。

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7. Identify any global extrema.
::7. 查明任何全球极端。

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8. Identify any local extrema.
::8. 标明任何局部端口。

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Use the graph below for 9-10.
::9-10使用下图。

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9. Identify any global extrema.
::9. 查明任何全球极端。

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10. Identify any local extrema.
::10. 查明任何局部端口。

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11. Explain the difference between a global maximum and a local maximum.
::11. 解释全球最高比率与地方最高比率之间的差别。

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12. Draw an example of a graph with a global minimum and a local maximum, but no global maximum.
::12. 绘制一个具有全球最低值和当地最高值但无全球最高值的图表的例子。

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13. Draw an example of a graph with local maximums and minimums, but no global extrema.
::13. 绘制一个带有本地最大值和最小值但无全球外形的图形示例。

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14. Use your graphing calculator to identify and categorize the extrema of:
::14. 使用您的图形计算计算器来识别和分类:

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       $\begin{array}{r}f(x)=\frac{1}{2}{x}^4+2x^3-6.5x^2-20x+24.\end{array}$
::f(x) = 12x4+2x3- 6.5x2-20x+24。

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15. Use your graphing calculator to identify and categorize the extrema of:
::15. 使用您的图形计算计算器来识别和分类:

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      $\begin{array}{r}g(x)=-x^4+2x^3+4x^2-2x-3.\end{array}$
::g(x)x4+2x3+4x2-2x-3。

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Review (Answers)
::回顾(答复)

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Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。