, (4)
(). f(x) , F(x, x0) .
.
4. f(x) (; b) -
,
, f (x) (; b), , f (x) (; b).
5 ( ). f (x) (; b) , - 1 < x 2 < x 3 (; b).
.
, .
5.5.4.
f(x) (; b) Î (; b). , , .
f(x) ( ) (; b), - d,c < d,
( ) (; b).
f(x) (; b), .
. 2 , f (x) - Î (; b) :
, f (x) . ♦
5.5.5.
1. (a; b)f(x) (), , f¢(x) ( ).
.
● f¢ (x) , . f (x) . ●
.
● , f¢ (x). , , , , < 0.
, f¢ (x) (0; + ¥) (¥; 0). f (x) (0; + ¥) (¥; 0).
2. f (x) [ a; b ].
1) f (x) () [ a; b ], , - Î [ a; b ].
f(x) ( ) [a; b], , , , (; d)Ì[a; b], .
. f (x) [ a; b ]. Î [ a; b ], f (x) ( ) [ a; b ].
5.5.4 , , . , f (x) , - () . . , = f (x) 0
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. (1)
, (1) (a; b), Î (a; b)
(2)
(1) f (x) (a; b), Î (a; b), ¹ 0,
. (3)
(2) (3) . .
3. f(x) () (a; b), , - f(x) (a; b) ( ) .
4. f(x) ( ) (a; b), , - f(x) (a; b) () f (x).
.
. f (x) () (a; b), - f (x) (a; b) ( ) (. 5.27).
. f (x) ( ) (a; b), - f (x) (a; b) () f (x).
. 5.27
. . f¢ (x). f (x) , f¢ (x) . f (x) = f (x), ( , ), f (x) = f (x), ( , ).
, .
●
( ¥; + ¥), , .
.
● ( 1; + ¥).
1 = 0 .
, (1; 0), . , ( 1; 0) , .
5.5.6.
f (x) (; b) , , Î (; b), , .
.
= f (x), Î (; b), ,
< = f (x) , > = f (x) , < = f (x) , > .
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. 5.28
(; f (c)) (. 5.28).
. , = f (x) , , . f (x) , = f (x) , , ( , ).
.
.
< 0 ¢() , > 0 . < 0 , > 0 ,
= 0 . , = 0 . (. 5.29).
. 5.29
5.5.7.
1 ( ). f (x) f² () . = f (x) f² () = 0.
. .
f² () ¹ 0. f² () f² () ¹ 0, , f² () f² (). f (x) ( f² () > 0), ( f² () < 0) = = f (x). . ¨
= f (x) .
2. f (x) < f¢ () , > , , < f¢ () , > , = = f (x).
3. f(x) f² () < 0 < , f² () > 0 > , , f² () > 0 < , f² () < 0 > , .
.
:
; ,
, ;
, ;
, .
,
. 5.30.
4. f (x) n - , f (n)() ,
f² () = f¢² () = = f (n 1)() = 0, f (n)() ¹ 0. ,
n ³ 3 ,
= f (x).
.
, , 4 = 0 . , ( ).
˲ ֲ
.
1- . .
2- . .
3- . .
5.6.1.
() , , , , , .
, , , , :
) , ;
) f (x) , f (x) = 0 ;
) f (x) j(x). f (j(x)) , j(x) , f (j(x)) . , f (x) = 4cos x R.
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) ;
) f (x) j(x) , , f (j(x)) j(x) , f (j(x)) ;
) f (x) , j(x) , , f (x) , j(x) , f (j(x)) j(x) , f (j(x)) . , f (x) = 2 x R, f (x) = 2cos x R, cos x, , cos x.
5.6.2.
1. f(x) [a; b] . f(x) [a; b], , .
. . , , .
. f (x) [ a; b ] . . ³ , f (x) [ 0; ], 0 < x [ ; 0], < x 0.
,
0 . , , , , Î[ a; b ], f (x) , f (x 0) [ a; b ].
5.6.3.
. f (x) (a; b) 0Î(a; b). , f (x) x 0, x 0, f (x) < f (x 0) < x 0, > x 0
f (x) > f (x 0).
f (x) 0Î(a; b), , f (x) > f (x 0) < x 0, f (x) < f (x 0) > x 0.
2 ( ). f(x) 0Î(a; b) f¢(x0) > 0 (f¢(x0) < 0), f(x) () 0.
. , f¢ (x 0) > 0. , 0, ¹ 0. , f (x) < f (x 0) < x 0, f (x) x 0. , f¢ (x 0) < 0.