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(9)

) # # $ %## F 0 [l, 1] , & 2, * & * ' * 6 * + ) x2 Î (l, 1)

/, )

(10)

. *

0 < x1 < l < x2 < 1,

x1x1 + (1 - x1)x2 ³ x2x1 + (1 - x2)x2,

# #0 * # f¢ * (, )

f(x1x1 + (1 - x1)x2) ³ f(x2x1 + (1 - x2)x2).

2 / * ( + * + x1 - x2 < 0, #0 (9) # (10) +!+ (# + * +

* 1 !" 3 0 + #' #+ (# + * +

F(l) £ lF(1) + (1 - l)F(0),

1 +#+ (3). ( + , $ %#/ f +! [a, b], ) 0 )#+ ( 0 * + ! 1.

1. ! % # f %% # [a, b]. - % #

, x l Î [a, b] f¢¢ (x) ³ 0.

. + 2( # * ( #0 &, ) #0+ ( / f¢ (+ 2(! (#$$ %# ' $ %## f 3!+ &( #

&(, &( f¢¢ (x) ³ 0. * ) (#$$ %# !" $ %#' ( (# ( & & #) * # %#

+! * #.

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4. % +

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& #) * ( # *+ '* + $ %##, 0 ) * 0!+ + & *. # #) * +! 2 #

( #/ + & * # *+ ( */ + # ( #/ 6 * 0 ' + 0 + * + #+ 2 !'. ) +#( , ) # * ' 1 # * (* +# 1 # #' + & * # (#$$ %# ' $ %## 0 )

#+ 2 !' " * # f¢ # # * + ( #+ * + * + f¢¢ (x) £ 0 ( / (+ 2(! (#$$ %# ' $ %## f.

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* * + , ) #0+ / $ %#/ + !" ) " 2 / " #* # # *+ & + ( #/, + ) /

" ( +! * # + & * # # # 3 .

#*. 11. ) .

( # 7. ! % # f [a, b] x0 Î (a, b). ) % # f [a, x0] [x0, b] , x0 % # f.

5. !

1 # * (* +# 1 ( 3!* 7 # 0 ( )# 3" (# * +## ) # .

2. (3" (# * +# ) # ). ! % # f %% # [a, b]

%% # x0 Î (a, b). ) x0 % # f, f¢¢ (x0) = 0.

. *# ! 1 #0+ ( / f¢ / " * # # " ( ) 0 ) x0. .1 f¢ (x) - f¢ (x0) / 0 + * * # ) # x0, * ( + ,

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# + 1 ' * * # 0 ! 0 # # x < x0 # x > x0. - )#, + ! 2( * 3 ' 0 ) #/

#

& # (# +!" 0 +, *

) ( 0!+ 2.

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( # 1. / # F % # f (a, b), F %% # x Î (a, b)

F¢ (x) = f(x).

) +#( , ) / # + 3 0 ' /+ / */ 3 ! 7 # #0+ ( ', %# !* #/

+ 3 0 ' 0 + * # # * * + # f® F # # F¢® F, * ( '* +#, 3

(#$$ %# + #.

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1. ) % # f F (a, b), % # f (a, b)

# 1 # + +! / */ + * + F¢ (x) = f(x). &( ( / 3 & ( '* +# & )#* C 2 * + ( #+ + * + (F + C)¢ (x) = F¢ (x) = f(x), ( 0!+, ) $ %#/ F + C 2 /+ / */ + 3 0 ' $ %## f (a, b).

2, ) 2 * + + 3 0 !" $ %## f + #*) !+ */ $ %#/ # F + C. .( 2#, )

H¢ (x) = F¢ (x) - G¢ (x) = f(x) - f(x) = 0.

!3 #0+ ) # x1 # x2 (a, b), x1 < x2. .# # , & 2 $ %## H (a, b), * & * '

'( */ ) c Î (x1, x2) /, ) * + ( #+ $

H(x2) - H(x1) = H¢ (c)(x2 - x1).

. * H¢ (c) = 0, # H(x2) - H(x1) = 0. 8 0!+, ) + +* " ) " # + (a, b) $ %#/ H # # (# +! 0 ) #/, * H(x) = C ( / & ( '* +# & )#* C 2( * + (a, b). 0+ 6 /*

$ %#/ F # G, #" (# $

G(x) = F(x) + C, x Î (a, b),

/ 0 )#+ ( 0 * + ! 1.

# 2 ** 2( #/ # * + #+, ) 1 * + ( #+ # + # ## 0 [a, b].

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2. !

( # 2. ! % # f [a, b]. " % # f

[a, b] % # f [a, b]

f (x)dx.

1 0!+, ) * # F - ( #0 + 3 0 !" $ %## f [a, b], ( !' # & /+ / */2 * + $ %#' F + C, &( C - #0+ ( '* +# )#*.

4 0 ! /+ / */ / # ( & # &.

( # 3. ! % # f F [a, b]. "

F(b) - F(a)

% # f [a, b]

b

f (x)dx.

a

( # 3 + * !*, ) 0 +#*# +!3 + 3 0 '. '* +# , * $ %#/ G

2 /+ / */ + 3 0 ' $ %## f [a, b]. & * 1 G = F + C, &( C - ( '* +# )#*. &(

G(b) - G(a) = (F(b) + C) - (F(a) + C) = F(b) - F(a),

) * + #+ * ( #/ 3.

. 2, ) ( !' # & 3 ( *+ '* + # ' * #.

2. ) % # f g [a, b] , % # af, a Î —, f + g

[a, b] %

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, %#/ 19 2. ( !' # ( !' # & !

( f + g )(x)dx = f (x)dx + g (x)dx.

a a a

. .* $ %#/ f # + 3 0 F, $ %#/ g # + 3 0 G 0 [a, b]. 8 0 )#, ) F # G (#$$ %# ! [a, b] # 1 0 * + ( #+! $!

F¢ (x) = f(x), G¢ (x) = g(x).

( + , $ %## aF # F + G 2 (#$$ %# ! [a, b] # * + ( #+! $!

(aF)¢ (x) = aF¢ (x) = af(x), (F + G)¢ (x) = F¢ (x) + G¢ (x) = f(x) + g(x).

8 0 ), ) $ %#/ aF /+ / */ + 3 0 ' $ %## af, $ %#/ F + G /+ / */ + 3 0 ' $ %## f + g # * + ( #+! $!

b

( f + g )(x)dx = (F + G)(b) - (F + G)(a) = F (b) - F (a) + G(b) - G(a) =

! 0 )#+ ( 0 * + ! 2.

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( 6 / +! 2 *+ '* + ((# #+ * # ( & # &.

3. ) % # f [a, b], c Î (a, b) % # f

[a, c] [c, b] %

) 0 )#+ ( 0 * + ! 3.

( # 3 ** #+ ( !' # & + * ), &( + 0 # & #2 #' ( * &

7 + " & (. ( * $ ( #/ 3 ( * * + ! * ( 6 * & 7 #: a > b

% # f, [b, a], ,

b a

f (x)dx = - f (x)dx,

a b

a = b ,

a

f (x)dx = 0.

a

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/ " 2( #/ + 3 0 !" 1 !" $ %#' ! * & * + ' 3 #% ' #0+ ( !", * (

)# + 3 /(:

(xn+1 )' = (n +1)xn xn dx = xn+1 + C, n ¹ -1; n +1

(ex )'= ex ex dx = ex ;

(log x)' = 1x dxx = log x + C;

(sin x)'= cos x cos xdx = sin x + C;

(cos x)'= -sin x sin xdx = -cos x + C;

189