( ) , .
1. . f(x) [a, b], (, b) f(a) = f(b), (, b) = x, f `( x) = 0. f(a) = f(b) = 0 ( ), , . : [a, b] , , b, x (a < x < b) , (tg a = f `(x) = 0).
2. ( ): f(x) [a, b] (a, b), = x, f(b) f(a) = (b a) f `( x).
: = f(x) x (a < x < b) , .
3. ( ): f(x) j(x) [a, b] (a, b), j`(x) ¹ 0, = x (a < x < b) , . , .
1. f(x) j(x) [a, b] ..
f(a) = j(a) = 0; , , , . 0/0 (, , ), : ) , f(x) j(x) = , ; ) f `(a) = j`(a) = 0, f `(x) j`(x) , , , ; ) ( ) .
.
2. f(x) j(x) ¹ , , . ,
(3.28).
. : ) = ¥; ) ¥.
, . , 0 × ¥ ¥ ¥ ; 00, ¥0, 1¥ .
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: 1. (∞-∞): ( , ). = (; ) =
2. 00: . . ( ∞0) = (; ) = (; ) = ; ; ℮0=1 .. ;
. y = f(x) . () : Pn(x a) = C0 + C1(x a) + C2(x a)2 + + Cn(x a)n (1) = f(). y = f(x) (n + 1) , i : , , .. . :
f()= C0; f `(a) = C1; f ``(a) = 2C2 = C2 ×2! ; f(n)(a) = Cn × n!
n! = n(n 2)(n 3) (n k) 3 × 2 × 1 ( n! n ). (3.29).
(1) : (3.30)
, = , f() n(x) . Rn(x) = f(x) Pn(x) :
(3.31)
Rn(x) . , , n(x) f(x). Rn(x)
, a < x < x (3.32).
( ). x : x = + q( ), 0 < q < 1 (3.32)
(3.32`)
(, , . Rn n , ).
(3.31), . = 0 (3.31`)
, 0 < q < 1 . , . :
(3.33)
(3.34)
(3.35)
. , , .
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1.17. , = 0, = 8? , x?
1) : x = 2; 3) : x = 4;
2) ; 4) : x = 6.
1.18. , (0; 0), (3; 18)?
1) (3; 1); 2) D(2; ); 3) (0; ); 4) F (; 0).
1.19. =
1) ; 2) ; 3) ; 4) .
1.20. =
1) 1; 2) 2; 3) 2; 4) 1.
1.21.
,
1) (1; 2; 3; 4); 2) (1; -1; 1; -1); 3) (1; -2; 3; -4); 4) (-1; 1; -1; 1).