4.1 Maximum and Minimum
Values
안수형
hera2021@ewha.ac.kr
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Maximum and Minimum Values
In general, we use the following definition.
An absolute maximum or minimum is sometimes called a
global maximum or minimum.
The maximum and minimum values of f are called extreme
values of f.
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Maximum and Minimum Values
In general, we have the following definition.
In Definition 2 (and elsewhere), if we say that something is
true near c, we mean that it is true on some open interval
containing c.
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Maximum and Minimum Values
Figure 2 shows the graph of a function f with absolute
maximum at d and absolute minimum at a.
Figure 2
Note that (d, f(d)) is the highest point on the graph and
(a, f(a)) is the lowest point.
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Example 4
The graph of the function
is shown in Figure 6.
Figure 6
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Maximum and Minimum Values
We have seen that some functions have
extreme values, whereas others do not.
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Maximum and Minimum Values
The following theorem gives conditions under which a
function is guaranteed to possess extreme values.
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Maximum and Minimum Values
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Maximum and Minimum Values
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Example 5
Find the critical numbers of
Solution:
The Product Rule gives
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Example 5 – Solution
cont’d
[The same result could be obtained by first writing
.]
Therefore
that is,
and f (x) does not exist when x = 0.
Thus the critical numbers are
and 0.
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Maximum and Minimum Values
In terms of critical numbers, Fermat’s Theorem can be
rephrased as follows (compare Definition 6 with Theorem 4):
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Maximum and Minimum Values
Thus the following three-step procedure always works.
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Example 6
Find the absolute maximum and minimum values of the
function
Solution:
Since f is continuous on
Interval Method:
we can use the Closed
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Example 6 – Solution
cont’d
Since f (x) exists for all x, the only critical numbers of f
occur when f (x) = 0, that is, x = 0 or x = 2.
Notice that each of these critical numbers lies in the interval
The values of f at these critical numbers are
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Example 6 – Solution
cont’d
The values of f at the endpoints of the interval are
Comparing these four numbers, we see that the absolute
maximum value is f(4) = 17 and the absolute minimum
value is f(2) = –3.
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