, , , . , .
, f (x) (. .5) , , [a,b], . . , "", (a,f(a)) (b,f(b)). , (1) x 0 , (a,f(a)) (b,f(b)) c . :
(3) |
x = x 0, y =0 : f(a)×f( x 0 ) f(b)×f( x 0 ), , 0, [ a, x 0] [ x 0 ,b ]. . , ( )
|b-a| <e. , , ( b).
. , f (x) 3 (. . 6). , , f (x) , x 0. :
) ;
) .
, f (x 0) × f '' (x) > 0,
[a,b],
.
(.7), [a,b], .. , " x Î[a,b]. , f ' (x) > 0 f '' (x) > 0, (.7 )), f ' (x) < 0 f '' (x) < 0, (.7 )). b , x 0=a. , , x 1 - (3) x. y = 0,
x 1 .
a) )
. 7. ,
, A1(x 1, f (x 1)) B (b,f(b)), :
:
(4) |
, , .. ," x Î[a,b] (.8). , f ' (x) < 0 f '' (x) > 0, (.8 )),
f ' (x) > 0 f '' (x) < 0, (.8 )). x = v [a,b] ( , .8) (5):
|
|
(5) |
a) )
. 8. ,
f (x) : [a,b] , . (4) , . , (5).
, x * e. x *= xn. - .2.
m | f ' (x)| [a, b], , :
e - .