( ).
f(x)=0, f(x) , .
f(x) - [a,b], , , b , . , , .
, ( ) " ". , , , 10 210=1024 . ε :
1. f(a) f(b).
2. = ( + b) / 2. f(c).
3. sign(f(c))=sign(f(a)), ; b ( sign(f(c)) ).
4. b - a >ε, 2; , . f(x) = 0.
, .
. ε=0,1 x3+3x2-3=0, [-3,-2].
.
:
n | an | bn | f(an) | f(bn) | cn | f(cn) | (bn-an)/2 |
-3 | -2 | -3 | -2,5 | 0,125 | 0.5 | ||
-3 | -2,5 | -3 | 0,125 | -2,75 | -1,11 | 0.25 | |
-2,75 | -2,5 | -1,11 | 0,125 | -2,625 | -0,42 | 0.125 | |
-2,625 | -2,5 | -0,42 | 0,125 | -2,5625 | -0,129 | 0.0625 |
, A=-2,56250,0625. , =-2,60,1.
( ).
:
f(x) [a, b]
f(a) f(b) : .. f(a)*f(b)<0;
f' (x) f'' (x) [, b].
, 1.
1. f(a)f"(x)<0, , .1 1. ' (.1)- '. , ,
(1)
, , , - f(x) 0x, D - 0x MM'. =0 (1), a1:
(2)
|
|
(a1, f(a1)) (b, f(b)) a2. , . an
(3)
2. f(a)f"(x)>0 (. .1 1) b,
(4)
, f(x) [a,b]. , - , xn - , ( ),
(5)
xn.
(2)
(6)
m f(x) [a,b], ..
(7)
m f(x).
. x3 -2x2 -4x-7=0 [3,4], 0,01.
. f(x).
f(3)=-10<0< f(4) = 9>0. , f'(x)=3x2- 4x-4, f''(x) = 6x- 4 [3,4
] .
, f"(x) , .. f"(x)>0, f'(x) . f'(3) = 11>0, [3,4] 11. , m= f'(3) = 11.
.. f(3)f"(x)<0, (3) (6).
, ,
.
, . . : εn=0,004<0,01. =3,6300,01.
. 2.2 ǹ1.