James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 4.1 Maximum and Minimum Values 281
The Closed Interval Method To find the absolute maximum and minimum values
of a continuous function f on a closed interval fa, bg:
1. Find the values of f at the critical numbers of f in sa, bd.
2. Find the values of f at the endpoints of the interval.
3. The largest of the values from Steps 1 and 2 is the absolute maximum value;
the smallest of these values is the absolute minimum value.
Example 8 Find the absolute maximum and minimum values of the function
f sxd − x 3 2 3x 2 1 1 2 1 2 < x < 4
SOLUtion Since f is continuous on f2 1 2 , 4g, we can use the Closed Interval Method:
y
20
15
10
5
_1
0
_5
FIGURE 15
8
0
_1
FIGURE 16
y=˛-3≈+1
(4, 17)
1 2
3 4 x
(2, _3)
2π
f sxd − x 3 2 3x 2 1 1
f 9sxd − 3x 2 2 6x − 3xsx 2 2d
Since f 9sxd exists for all x, the only critical numbers of f occur when f 9sxd − 0, that is,
x − 0 or x − 2. Notice that each of these critical numbers lies in the interval s2 1 2 , 4d.
The values of f at these critical numbers are
f s0d − 1 f s2d − 23
The values of f at the endpoints of the interval are
f s2 1 2 d − 1 8
f s4d − 17
Comparing these four numbers, we see that the absolute maximum value is f s4d − 17
and the absolute minimum value is f s2d − 23.
Note that in this example the absolute maximum occurs at an endpoint, whereas the
absolute minimum occurs at a critical number. The graph of f is sketched in Figure 15.
n
If you have a graphing calculator or a computer with graphing software, it is possible
to estimate maximum and minimum values very easily. But, as the next example shows,
calculus is needed to find the exact values.
Example 9
(a) Use a graphing device to estimate the absolute minimum and maximum values of
the function f sxd − x 2 2 sin x, 0 < x < 2.
(b) Use calculus to find the exact minimum and maximum values.
SOLUtion
(a) Figure 16 shows a graph of f in the viewing rectangle f0, 2g by f21, 8g. By
moving the cursor close to the maximum point, we see that the y-coordinates don’t
change very much in the vicinity of the maximum. The absolute maximum value is
about 6.97 and it occurs when x < 5.2. Similarly, by moving the cursor close to the
minimum point, we see that the absolute minimum value is about 20.68 and it occurs
when x < 1.0. It is possible to get more accurate estimates by zooming in toward the
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