f (x)=0 [ a; b ], f (x) f ''(x) Î [ a; b ].
: xn xn1 :
, x0
, .
:
1) x*;
2) x0;
3) , x* , .
( 3). x0 X y = f(x). y = f(x) X: x1. x1 X y = f(x), x2.
, , , , x*. 4 , , , , . , .
3 y=f(x) , f'(x) f''(x) ( ). ? 5 , . y= f(x) ( ) ( ). x1, x2, r, x3, f(x).
6 , f(x) . , , [a; b ] ( ). x0 , b, , 3, . , a, a ( b), . [a; b ], , .
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, x0=b x0 , .
, , ( 7):
f(x) = 0 [a;b] ( f(x) [a;b] ), f'(x) f''(x) [a;b]. , 0 Î [ a; b ] , f'(x0) ∙ f''(x0) > 0, ,
.
cos (x) 3 x + 1 = 0, [0;1] ( 8).
f'(x) = sin (x) 3 < 0 x. f''(x) = cos (x) < 0 [0;1]. , , f(x0) < 0. : x0 = 1.
x0 = 1 ( 9):
x1 = x0 (cos(x0) 3x0 + 1)/(-sin(x0) 3) = 1 (0.54 3 + 1)/(-0.84 - 3) = 0.62
x2 = x1 (cos(x1) 3x1 + 1)/(-sin(x1) 3) = 0.62 (0.814 1.86 + 1)/(-0.581 - 3) = =0.607
x3 = x2 (cos(x2) 3x2 + 1)/(-sin(x2) 3)=0.607 (0.821 1.821 + 1)/(-0.57 - 3)= =0.607
, .
( 10), n
m 1 |f'(x)|
M2 | f (x)|
, ε, :
, . , :
m 1 |f'(x)|
, , .
11 . x, E, m 1 M2. x. , f (x) F(x), f (x) F1(x).
f (x)=0 [ a; b ], f (x) f ''(x) Î [ a; b ]. .
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12 , f(x) f (x) . A B :
x0 x0 = a, x 1 O x, y = 0. :
x1 A1 y =f(x), . Ox x2:
n :
, f'(x) > 0, f''(x) < 0 ( 13). , y=f(x), a b, :
x0 = b, :
, f'(x) < 0 [a;b].
, , , , . ( 14):
[a;b], : f(x)∙ f (x)>0; .
cos (x) 3 x + 1 = 0, [0;1]. . f''(x)<0, b=1, f(b)=-1.4597<0, x0=a=0,
( 15):
x 1 = x 0 (cos (x 0) 3 x 0 + 1)/(-1.4597- cos (x 0) 3 x 0 + 1)∙(1- x 0) =
= 0 2/(-1.4597-2)∙1 = 0.5781
x2 = x1 (cos (x1) 3x1 + 1)/(-1.4597- cos (x1) 3x1 + 1)∙(1- x1) =
= 0.5781 0.1028/(-1.4597-0.1028)∙(1-0.5781) = 0.6059
x3 = x2 (cos (x2) 3x2 + 1)/(-1.4597- cos (x2) 3x2 + 1)∙(1- x2) =
= 0.6059 0.0043/(-1.4597-0.0043)∙(1-0. 6059) = 0.6070
, , , .
( 16), n
m 1 |f'(x)|
M1 | f (x)|
, ε, :
17 . x, c, E, m 1 M1. x. , f (x) F(x).
F(c) fc. F(x), , fx.
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, . , , .