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5
x*
x=3
0
f(x)
−5
−10
−15
−20
0
0.5
1
1.5
2
2.5
3
3.5
x
e x
Fig. 4.1
The graph of f.x/
D
2
C
x
We note that f.x 0 / D f.0/ > 0 and f.x 1 / D f.3/ < 0. Next, we define the mean
value
1
2 .x 0 C x 1 / D
3
2 :
x 2 D
(4.36)
Since we are looking for a point where f is zero, our next step thus depends on
the value of f.x 2 /. Obviously, if f.x 2 / is zero, we just put x D x 2 , and our search
is completed. But f.x 2 / is not zero. In fact,
f.x 2 / D f 3
2
D 2 C 3=2 e 3=2 <0;
(4.37)
see Fig. 4.2 .
Since f.x 0 />0and f.x 2 /<0, we know that x 0 <x <x 2 .Next,wedefine
1
2 .x 0 C x 2 / D
3
4 :
x 3 D
 
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