Hms 06 14-math_kat
367
ΕΛΛΗΝΙΚΗ ΜΑΘΗΜΑΤΙΚΗ ΕΤΑΙΡΕΙΑ ΕΠΑΝΑΛΗΠΤΙΚΑ ΘΕΜΑΤΑ Γ΄ΛΥΚΕΙΟΥ 2006 1 ΘΕΜΑ 1 Έστω f (z) z iz = − , z ∈ . α) Να λύσετε την εξίσωση : f (z) 2 i = − . β) Αν f(z) 2 = να βρείτε το z . γ) Αν z 1 = να δείξετε ότι ο γεωμετρικός τόπος των εικόνων του w=f(z) είναι κύκλος που διέρχεται από την αρχή των αξόνων. ΛΥΣΗ α) f (z) 2 i z iz 2 i = − ⇔ − = − . Αν z x yi = + τότε 2 2 2 2 x y i(x yi) 2 i ( x y y) xi 2 i + − − = − ⇔ + − − = − ⇔ 2 2 x y y 2 + − = 2 1 y y 2 + = + 2 2 1 y y 4y 4, y 2 + = + + ≥− ⇔ ⇔ x 1 = x 1 = x 1 = 3 y 4 x 1 ⎧ =− ⎪ ⇔ ⎨ ⎪ = ⎩ . Άρα 3 z 1 i 4 = − . β) f(z) z iz z iz (1 i) z = − = − = − οπότε f(z) 1 iz 2z = − = . Άρα 2z 2 z 1 = ⇔ = . γ) Αν z 1 = τότε: w 1 iz iz w 1 = − ⇔− = − οπότε w 1 iz w 1 z z 1 − =− ⇔ − = = = άρα w 1 1 − = Η τελευταία σχέση εκφράζει μια εξίσωση κύκλου που επαληθεύεται για w=0. ΘΕΜΑ 2 Δίνεται η εξίσωση : ( ) zz 4Re 1 2i z 4 0 + − + = ⎡ ⎤ ⎣ ⎦ (I). α) Να δείξετε ότι η εξίσωση αυτή έχει άπειρες λύσεις στο σύνολο των μιγαδικών. β) Αν 1 2 z ,z είναι δύο λύσεις της παραπάνω εξίσωσης, να δείξετε ότι: 1 2 z z 8 − ≤ . γ) Αν 1 2 t ,t είναι αντίστοιχα οι τιμές των μιγαδικών 1 z και 2 z (του ερωτ. β) για τις οποίες η παράσταση 1 2 z z − γίνεται μέγιστη, να δείξετε ότι: 2v v 4v 1 v 1 2 1 2 t t 10(t t) 2 5 + + + − = ⋅ για κάθε * ν ∈ .
Transcript of Hms 06 14-math_kat
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1
f (z) z iz= , z^ . ) : f (z) 2 i= . ) f ( z ) 2= z . ) z 1= w=f(z)
.
) f (z) 2 i z iz 2 i= = . z x yi= + 2 2 2 2x y i(x yi) 2 i ( x y y) xi 2 i+ = + =
2 2x y y 2+ = 21 y y 2+ = + 2 21 y y 4y 4, y 2+ = + + x 1= x 1= x 1=
3y4
x 1
= =. 3z 1 i
4= .
) f ( z ) z i z z i z (1 i) z= = = f ( z ) 1 i z 2 z= = . 2 z 2 z 1= = . ) z 1= :
w 1 iz iz w 1= = w 1 iz w 1 z z 1 = = = = w 1 1 =
w=0.
2
: ( )zz 4Re 1 2i z 4 0+ + = (I). )
. ) 1 2z , z , :
1 2z z 8 . ) 1 2t , t 1z 2z ( . )
1 2z z , : 2v v 4v 1 v1 2 1 2t t 10(t t ) 2 5
++ + = *` .
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) z=x+yi, (x,y )\ () : 2 2 2 2x y 4x 8y 4 0 (x 2) (y 4) 16+ + + + = + + + = . (I) ,
C : 2 2(x 2) (y 4) 16+ + + = . ) 1 2z , z (I)
1 1M (z ) 2 2M (z ) C . :
1 2 1 2z z (M M ) 2 2 4 8 = = = . ) 1 2z z
1 2M , C , . , 1t 2t .
31 2 1 2t t max z z 8 2 = = = .
21 2t t OA OB 2 OK 2 20 2 5+ = + = = =
JJJG JJJG JJJG.
:
2v v 4v v 4v v 4v v 4v 1 v
1 2 1 2t t 10(t t ) 2 5 2 5 2 2 5 2 5++ + = + = = .
3
f : \ \ : (I) 3 2 3f (x) 2x f (x) 3 x+ = x\ .
x 0
f (x)lim x
= : ) =1.
) : i) x 0
f (x)limx
, ii) x 0
f (f(x))limx
iii) 2
2x 1
f (x x)limx 3x 2
+ .
) x 0 3x (I) : 3 3f (x) f (x) x( ) 2 3( )x x x
+ = : 3 2 3 1+ = = .
) 0
i) f (x) f (x) xx x x
= x 0
f (x)limx
y=x y 0
f (y)lim 1y
= .
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ii) f (f (x)) f (f (x)) f (x)x f (x) x
= x 0 x 0
f(x)limf(x) lim(x ) 0 1 0x
= = =
x 0
f (f(x))limx
y=f(x) y 0
f (y)lim 1y
= .
iii)2 2 2 2 2
2 2 2 2 2
f (x x) f (x x) x x f (x x) x(x 1) f (x x) xx 3x 2 x x x 3x 2 x x (x 1)(x 2) x x x 2
= = = + +
1 : 2
2x 1
f (x x)lim 1 ( 1) 1x 3x 2
= = + .
4
f : \ \ , . x 1
f (x)lim 1x 1
=
) : x2
f (x)limx
) : f(1)=0
) k, f (x) ,x 1
g(x) x 1k, x=1
=
\ . ) k=1 g
(): y=2x 0x (0,1) .
) 2
2 2
f (x) f (x) f (x) f (x) xx x
x x 1 x x-1 1 x= = = +
y=x y 1x2
f (x) f (y)lim lim 1x-1 y-1
= = . x2
f (x)lim 1 0 0x
= = .
) f 1 : f(1)= x 1 x 1
f (x)limf (x) lim(x 1) 0 1 0x 1
= = = .
) g(1)= x 1 x 1
f (x)limg(x) lim 1x 1
= = k=1. ) Bolzano h(x)=g(x)-2x [0,1]. : h(0)=g(0)=-f(0) f \ , : f (0) f (1) 0< = . h(0) 0> . : h(1)=g(1)-2=-1 0< .
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f : \ \ : (I) f (x) x 1 x\ . ) :
x
f (x)lim 1x+
= . ) :
i) vxf (x)lim ,x+
*v` , ii) x 0
1lim[x f ( )]x+
, iii) 2 2
2x
x x 2 xlim [ f ( )]x x 1+ + .
) x 0> : f (x) 1 1 f (x) 1f (x) x 1 1 1 1x x x x x
+
: x
f (x)lim 1x+
= .
) i) : >1 v v 1x xf (x) f (x) 1lim lim [ ] 1 0 0x x x + +
= = =
=1 vx xf (x) f (x)lim lim 1x x+ +
= = .
ii) x 0 x 0
1f ( )1 xlim[x f ( )] lim 1xx
+ + = 1u
x=
u
f (u)lim 1u+
= .
iii)
2
2 2 2
22 2
xf ( )x x 2 x (x 2)(x 1) x x 1f ( ) f ( ) (x 2)xx x 1 x x 1
x 1
+ + = = + ++
(1)
2xu
x 1= + :
2
2x x
xf ( ) f (u)x 1lim lim 1x u
x 1+ +
+ = =+
(1) : 2 2
2x
x x 2 xlim [ f ( )]x x 1+ + = + .
6
f \ : (I) x xf ( )x 1 x 1
= x 1 . ) :
x
xlim 0x 1+
= . ) f(1).
) 0 x ( , )-6 -2
: 0f(x ) 0= .
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) x>1 : xx 1 1 x 1
x-1 x-1 x 1 x 1 x 1 x 1=
: x
xlim 0x 1+
= .
) (I) : x
xlim f ( ) 0x 1+
= . xy
x 1=
x y 1
x0 lim f ( ) limf (y)x 1+
= = f 1 :
y 1f (1) limf (y) 0
= = .
) Bolzano f [ , ]-6 -2
.
- x=6
(I) : 1
32f ( ) 0-6 616
= = .
: (I)
x 1 , ) ) .
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f [0, )+ f(x)>0 x [0, ) + . f, xx, yy x=u uE(u) e f (u)= u 0 : i) xf (x) f (x) e + = x [0, ) + ii)
x 0
f (x) 1lim 0x = .
iii) f. iv) 1f (x) x
2 = .
i) f [0, )+ f(x)>0 x [0, ) + ,
fC , xx, yy x=u,
u
0
E(u) f (x)dx= u 0 u u0
f (x)dx e f (u)= (1). f [0, )+ , 0 [0, ) +
u
0
( f (x)dx) f (u) =
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u 0 u
u u u
0
( f (x)dx) (e f (u)) f (u) e f (u) f (u) f (u) e = = + = . x [0, ) + xf (x) f (x) e + = (2). (1) u=0 f(0)=1 (2) x=0 f (0) 0 = .
x 0 x 0
f (x) 1 f (x) f (0)lim lim f (0) 0x x 0 = = = ( f [0, )+
). ii) x [0, ) +
xex x x 2x x 2x1f (x) f (x) e f (x)e f (x)e e (f (x)e ) ( e )
2
+ = + = = x 2x1f (x)e e c
2= + , x 0 (3).
x=0 (3) 1 11 c c2 2
= + = x 0 x 2x x x1 1 1f (x)e e f (x) (e e )
2 2 2= + = + , x 0 .
iii) x [0, ) + x x1f (x) (e e )2
= , 1f (x) x2
=
x x x x1 1(e e ) x e e x2 2
= = (4). 0 0e e 0 0 0 = = , 0 . x xg(x) e e x= + , x 0 . x [0, ) +
x x xx
00
1g (x) e e x=(e 2) (2 x) 0e
>
= + + + >
g
[0, )+ , 0 . x -x1f (x) x e e x x=0
2 = = .
: 1 2
+ >0 xe 0> x
x
1e 2 0e
+ . 2-x>0 x 1 x.
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f \ f (x) 2007 > x\ . fC y=2006x+1 .
f(x) =2006x+1 \ . g(x)=f(x)-( 2006x+1), x\ . H g \
g (x)=f (x)- 2006>2007-2006=1>0 (1), x\ , g \ , g(x)=0 \ .
g .. [0,x] x>0 1 (0,x)
1 1g(x)-g(0)g ( )= g(x)=g ( ) x+g(0)>x+g(0)
x 0 (
x>0
1 1g ( )>1 g ( )x>x ). ( )
xlim x g(0)+ + = + xlim g(x)+ = + ,
(, )+ >0 g(x)>0 x (, ) + . g()>0 (, ) + . g .. [x,0] x
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) z Rez 0 , f (x) xe 2e x z e 2+ + x ( e, ) + .
) f ( t )e ( e, ) + , 0 ( e, ) +
xf ( t )
0
e dt ( e, ) + , f ( e, ) +
xf ( t ) f (x)
0
f (x) (1 e dt) f (x) e = + = f (x)
1f (x)e
= (1) f (x) f (x) f (x)e f (x) 1 [e ] (x) e x c = = = + , c\ (2).
x=0 f(0)=1, (2) f (0)e c c e= = .
f (x)e x e f (x) ln(x e)= + = + , x>-e.
) (1) f (x)1f (x) 0
e = > f
( e, ) + , 1-1. f . f y=f(x) x.
y y 1 yy f (x) y ln(x e) x e e x e e f (y) e e= = + + = = = , y\ . f 1f : \ \ 1 xf (x) e e = . ) y=0 ln(x e) 0 ln(x e) ln1 x e 1 x 1 e+ = + = + = = (1-e,0).
0 0 0 001 e
1-e 1-e 1-e 1-e
x()= ln(x e)dx x ln(x e)dx [x ln(x e)] x(ln(x e)) dx dx
x e + = + = + + = =+
0 0 0 0 0 0
1-e 1-e 1-e 1-e 1-e 1-e
x e e e 1 1( )dx (1 )dx e dx dx e (x e) dx dxx e x e x e x e x e+ = = = = + =+ + + + +
01 ee[ln x e ] [0 (1 e)] e 1 e 1= + = + = ..
) f (x) xg(x) e 2e x z e 2= + , x ( e, ) + . x ( e, ) + f (x) x f (x) xe 2e x z e 2 e 2e x z e 2 0 g(x) g(0)+ + + . g(x) g(0) x ( e, ) + , g 0x 0= . g
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( e, ) + f (x) xg (x) e f (x) 2e z = + 0x 0= (1)
f (0) 0 f (0)f (0)
1g (0) e f (0) 2e z g (0) e 2 z g (0) 3 ze
= + = + = . Fermat, g (0) 0 3 z 0 z 3 = = = . z (0,0) =3
2 2x y 9+ = . Rez 0
z qB AB 2 2x y 9+ = (0,-3). (3,0), (0,3).
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f 2 2f (x) 2xf (x) x f (2 x)= x\ f(2)=0. ) 2 2[f(x)-x] x [1 f (2 x)]= x\ . ) f (x) 1 x\ . ) fC xx , x=1 f f(1)=1.
) x\ 2 2 2 2f (x) 2xf (x) x f (2 x) f (x) 2xf (x) x f (2 x)= = 2 2 2 2 2 2f (x) 2xf (x) x x x f (2 x) [f (x) x] x [1 f (2 x)] + = = . ) x 0
2
2
[f (x) x]1 f (2 x) 0x = f (2 x) 1 *x\ .
f (2 0) f (2) 0 1 = = . f (2 x) 1 x\ . x 2-x f (2 (2 x)) 1 f (x) 1 x\ .
) x=1 2 2f (1) 2f (1) f (1) f (1) f (1) f (1)[f (1) 1] 0 f (1) 0= = = = f (1) 1= .
x=0 2f (0) 0 f (0) 0= = . f (0) 0= f(2)=0 f(1)=0 fC
xx . f(1)=1. f (x) f (1) x\ . f 1 x=1.
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f, g \ . f gD 1-1, (f g)(0) 0=D x\ g(x) x
0 0
f (t)dt (f g)(t)dt 1+ = D , : ) g(x)=-1 *x\ )
g(x) x
0
f (t)dt 1+
= , x\ .
) f \ , 0\ g \ ,
g(x)
0
f (t)dt \ . f gD \ , 0\
x
0
(f g)(t)dt D \ .
x\ g(x) x
0 0
( f (t)dt (f g)(t)dt) 0+ = D f (g(x))g (x) (f g)(x) 0 (f g)(x)[g (x) 1] 0 + = + =D D (1). f gD 1-1 (f g)(0) 0=D
(f g)(x) 0 (f g)(x) (f g)(0) x 0= = =D D D . (f g)(x) 0=D x=0. (1)
x=0 g (x) 1 0 + = , *x\ . )
g(x) x
0
h(x) f (t)dt+
= , x\ . (2) f \ , 0\ , g(x)+x \ , h \ h (x) f (g(x) x) (g(x) x) h (x) f (g(x) x) (g (x) 1) = + + = + + , x\ . x=0 h (0) f (g(0)) (g (0) 1) 0 (g (0) 1) 0 = + = + = . x 0 h (x) f (g(x) x) 0 0 = + = . h (x) 0 = x\ , h \ , h(x) c= , x\ . x=0
g(0)
0
f (t)dt 1= (2)
g(0)
0
h(0) f (t)dt c 1= = . x\ g(x) x
0
h(x) 1 f (t)dt 1+
= = .
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) : ) f (x) ln x= , x [1,e]
1f .
) 2e 1
x
1 0
ln xdx e dx e+ = . ) 2
xt
1
h(x) e dt= . : ) xh(x) e e , x [0, ) + .
) 1
0
1 eh(x)dx2= .
) ) x (1,e) 1f (x) 02x ln x
= > f [1,e] f [1,e], 1-1, . f f([1,e])=[f(1),f(e)]=[0,1]. y=f(x) 1 x e , 0 y 1
2 22 y 1 yy ln x ln x y x e f (y) e= = = = . 1f :[0,1] \ 21 xf (x) e = [ ]1f ( 0,1 ) [1,e] = . )
e
1
I ln xdx= ln x u= 2ux e= , 2 2u udx (e ) du dx 2ue du= = .
x=1 u ln1 0= = x=e u ln e 1= = . 2 2 2 2 2 2
1 1 1 1 1u u u 1 u u x
00 0 0 0 0
I u2ue du u(e ) du [ue ] u e du e e du e e dx = = = = = , 2
e 1x
1 0
ln xdx e dx e+ = 21 x0
e dx + 21 x0
e dx =e.
) ) 2x
x t x
1
(x)=h(x)-e e e dt e e+ = + , x \ . 2te \ , 0\ , 2
xt
1
e dt \ 2 2
xt x
1
( e dt) e = . \ . x \ 2x x (x)=e e .
2 2x x x x 2 (x)=0 e e 0 e e x x x(x 1) 0 x 0 = = = = = x 1= . 2 2x x x x 2 (x)>0 e e 0 e e x x x(x 1) 0 x 0 > > > > < x 1> .
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x - 0 1 + (x) + 0 - 0 + - (x) (0) 0
21
t
1
(1)= e dt e e 0 + = . [0, )+ 0x 1=
(1)=0 . x [0, ) + x x(x) (1) h(x)-e +e 0 h(x) e -e. )
2 21 1 1 1 1
1 x x0
0 0 0 0 0
1h(x)dx (x) h(x)dx [xh(x)] xh (x)dx h(1) xe dx 0 (e ) dx2
= = = = = 2x 10
1 1 1 e[e ] (e 1)2 2 2
= = = .
13
f (0, + ), f(x)>0 x>0, f(1)=1, f(2)=3
2 2
1 1
f (xt)dt f (x)dx . )
2x 2
x 1
1 f (u)du f (x)dxx
. )
f xx x=1 x=2.
) 2
1
I f (xt)dt= . xt=u
x>0 1xdt=du dt= dux
. t=1 u=x. t=2 u=2x.
2x 2x
x x
1 1I f (u) du f (u)dux x
= = : 2x 2
x 1
1 f (u)du f (x)dxx
. )
2x 2
x 1
1h(x) f (u)du f (x)dxx
= , x>0. x (0, ) +
2x 2 2x 2
x 1 x 1
1 1f (u)du f (x)dx f (u)du f (t)dt 0x x
h(x) 0 h(x) h(1) h 0x 1= .
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2x x 2
1 1 1
1 1h(x) f (u)du f (u)du f (x)dxx x
= (0, )+ : 2x x
2 21 1
1 1 1 1h (x) f (u)du f (2x) (2x) f (u)du f (x) 0x x x x
= + + x 2x
21 1
1 2 1h (x) ( f (u)du f (u)du) f (2x) f (x)x x x
= + . 0x 1=
2
1
h (1) f (u)du 2f (2) f (1) = + . .Fermat.
2 2
1 1
h (1) 0 f (u)du 2f (2) f (1) 0 f (u)du 2 f (2) f (1) = + = = 2 2
1 1
f (u)du 2 3 1 f (u)du 5..= =
14
) xe x x+1-x x x [0, ) + . ) x
x-xf(x)= ,e x x 0 . :
i) f. ii) , x xx-x=(e x)+(e x) x [0, ) + . iii)
f, xx x=0 x4
= .
) 1 x 0 x+1-x x . xe x x+1-x . xe x 1 + .
xh(x)=e x 1 , x [0, ) + . xh (x)=e 1 . h [0, )+ , h (x)>0 (0, )+ . h [0, )+ x 0
x xh(x) h(0) e x 1 0 e x 1 + .
) i) xx xx-xlim f (x) lime x+ +
= . x (0, ) +
x x
x-xx-xf (x)e x e x
= =
xx x x x
x-x x x 1 1 2 2 2e x xe x e x e x e x
+ + = = )
. , 2 2f (x)x x
x (0, ) + .
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x
2lim ( ) 0x+
= x
2lim 0x+
= xlim f (x) 0+ = ,
y=0 fC + . ii) x xx-x=(e x)+(e x) x [0, ) +
x x xx-x=(e x)+(e x) (+)e ( 1)x-(-1)x=0 + x [0, ) + . x=0 0(+)e ( 1)0-(-1)0=0 1 0 1 + + + = = . x
2= 2 2 (+)e ( 1) -(-1) =0 ( )e 0 0 1
2 2 + + = + = = .
x xx-x=-(e x)+(e x) , x [0, ) + . iii)
x x4 4 4 4
x x x x0 0 0 0
x-x x-x (e x) (e x)E f (x) dx dx dx [ ]dxe x e x e x e x
= = = = + =
x4x 4 4
0x0
(e x) 2[ 1 ]dx [ln(e x)] ln(e )e x 4 4 2
= + = = .
x [0, ]4
x x x-x 0 xe x 0 > .
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f \ f (x) 0 x
t
1
e f (t)dt x 1 x \ . : ) 1f (1)
e= .
) f (x) 0> x \ . ) f(x)=x (0,1). ) z
5z 2e f (1) i2
+ = .
) x
t
1
g(x) e f (t)dt x 1= + , x \ . x \ x x
t t
1 1
e f (t)dt x 1 e f (t)dt x 1 0 g(x) 0 g(x) g(1) + . g(x) g(1) x \ , g
0x 1= . te f (t) \ , 1\
xt
1
e f (t)dt \ . x+1 \ g \
xt
1
g (x) ( e f (t)dt x 1) = + xg (x) e f (x) 1 = . g 0x 1=
1g (1) e f (1) 1 g (1) ef (1) 1 = = . .Fermat, 1 1g (1) 0 e f (1) 1 0 f (1)
e = = = .
) f \ x \ , f . 1f (1) 0
e= > f (x) 0> x \ .
) h(x) f (x) x= , x [0,1] .
h [0,1] 1h(0) h(1) [f (0) 0] [f (1) 1] f (0) [f (1) 1] f (0) ( 1) 0
e = = = < f(0)>0
1 1 0e
< . .Bolzano h(x) 0 f (x) x= = (0,1) .
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) 5 1 5 5 5z 2e f (1) i z 2e i z 2 i z (2 i)
2 e 2 2 2 + = + = + = = .
z
(2,-1) 52
= 2 2 5(x 2) (y+1)4
+ = . (0,0) (2,-1).
K O
K O
y y -1 0 1
x x 2 0 2 = = = :
1y x y x2
= = .
2 2
1 : y x
25c: (x 2) (y+1)4
= + =.
2
2 2 2 2 21 5 2 x 5 (2 x) 5(x 2) ( x+1) (x 2) ( ) (x 2)2 4 2 4 4 4
+ = + = + = 2 25 5(x 2) (x 2) 1 x 2 1 x 3
4 4 = = = = x 1= .
x 3= 3y2
= x 1= 1y2
= .
13z 3 i2
= 2 2
13 3 5z 3 ( )2 2
= + =
21z 1 i2
= 2 22 1 5z 1 ( )2 2= + = .
16
2006
1f (x) 2,1+x
= + x \ . ) f . ) f (x) =
\ . )
x 1
xx
lim f (t)dt+
+ .
-
2006
17
) x \ 1 3
2006 2006 20062 22006
1 1f (x) ( 2) [(1+x ) 2] (1+x ) (1+x )21+x
= + = + = =
2005
2005
2006 3 2006 2006
1 1 1003x2006x2 (1 x ) (1 x ) 1 x
= = + + + .
f (x) 0 x 0 = = f (x) 0 x 0 > < ,
x - 0 + f + 0 - f 3
f ( ,0] f (x) 0 > ( ,0) , f
( ,0] . f [0, )+ f (x) 0 < (0, )+ , f
[0, )+ . f 0x 0= f (0) 3= . ) f.
2006x x
1lim f (x) lim ( 2) 21+x
= + = 2006x x
1lim f (x) lim ( 2) 21+x+ +
= + = .
x xf ( ) ( lim f (x),f (0)] ( lim f (x),f (0)] (2,3] += =\ .
: ( ,2] (3, ), + f (x) = , f( ) \ . 3= f (x) = x 0= . (2,3) f (x) = 2 ,
. ) f [x,x 1]+ x>0,
x t x 1 + f (x) f (t) f (x 1) + . f \
x 1 x 1 x 1 x 1 x 1 x 1
x x x x x x
f (x)dt f (t)dt f (x 1)dt f (x) dt f (t)dt f (x 1) dt+ + + + + +
+ +
x 1 x 1
x x
f (x)(x 1 x) f (t)dt f (x 1)(x 1 x) f (x) f (t)dt f (x 1),x 0+ +
+ + + + >
xlim f (x) 2+ = xlim f (x 1) 2+ + = ( x+1=u
x ulim f (x 1) lim f (u) 2+ ++ = = )
x 1
xx
lim f (t)dt 2+
+ = .
-
2006
18
17
f \ , f (x) 0 x \ , 1f(2005)=2
, f(2007)=3
f(1) f(2)=f(3) f(4) ) \ f()=1 ) 0x [1,2] 2 0f (x )= f(1)f(2) ) f .
) f [2005,2007]
1 =f(2005) f(2007)=32
1 f(2005) f(2007)
, (2005,2007) \ f()=1. ) 2g(x)=f (x)- f(1)f(2), x [1,2] . g [1,2] 22 2
0 0
g(1)g(2)=[f (1)- f(1)f(2)][f (2)- f(1)f(2)]=- f(1)f(2) [f(1)- f(2)] 0<
f \ f(1)f(2)>0 f(1)f(2)
-
2006
19
18
f 0, ( )f 0 2006= ( ) ( )4 1xf x x xx
+ = , 0 *x \ ) ) f \
) . ) ) ( )
xlim f x ( )xlim f x+
) ( )f x 0= \ .
) ) x 0 ( ) ( ) 3 x 1f x x x x
= , f *\ f 0 f \ . ) 0x 0=
( ) ( ) 3x 0 x 0
x 1lim f x lim x x x
= = .
3x 0
1lim(x ) 0x
= , x 0 3 31x xx
3 3 3
3 3
x 0 x 0
1x x xx
lim x lim x 0
= = 3
x 0
1lim(x ) 0x
= .
( ) ( )x 0 x 0
x xlim lim[ ] 1
x x = = = .
( )x 0lim f x f(0)= =2006 = , f 0.
)
( )x
xlim 0
x= ( )
( ) x 1x x
( ) ( )
x x
x x1 1 1 1x x x x x x
1 1lim lim 0x x
= = ( )
x
xlim 0
x=
3x
1lim x x
= 3
xlim x = x t 0
1lim lim(t)=1x
= . ( )
xlim f x = + , , .
-
2006
20
) + ( )
x
xlim 0
x+= ( ) 3
x
1lim (x )x+
= + 3xlim x+ = + ,
x
1lim 1x+
= . ( )
xlim f x+ = , + , >0, ( )f 0< .
f [,] ( ) ( )f f 0< Bolzano ( )f x 0= ( ), \ .
19
) f [0,] ( ) ( ) 0 0
f x dx f x dx= ) ( ) ( )f x ln 1 x= + , x 0,
4
) ( ) f x f x ln 24
+ = x 0,4
) f y 0= , x 0=
x4
= .
) x u = , du=-dx. x = u 0= x 0= u =
( ) ( ) ( ) ( ) 0 0 0 0
f -x dx f u du f u du f x dx= = = ) ( ) ( )f x ln 1 x= + , x 0,
4 .
x 0,4
0 x 0 x x 0 0 x4 4 4 4 4 4
. ) )
) f 0,4
x 1 x4f x ln 1 x ln 1 ln 14 4 1 x1 x4
= + = + = + + + ( )1 x 1 x 2ln ln ln 2 ln 1 x
1 x 1 x + + = = = + + +
( ) ( ) ( )f x f x ln 1 x ln 2 ln 1 x4
+ = + + +
-
2006
21
( ) f x f x ln 24
+ = ) ,
( ) ( ) 4 4
0 0
E f x dx ln 1 x dx= = +
. . 0 x 0 x 0 x 1 1 1 x 24 4
+ ln1 ln(1 x) ln2 0 ln(1 x) ln2 + +
( ) ( ) 4 4
0 0
E ln 1 x dx f x dx= + = ) :
( ) f x f x ln 2.4
+ =
( ) 4 4 4
0 0 0
f x dx f x dx ln2dx4
+ =
) = 4
( ) 4 4
0 0
f x dx f x dx4
=
( ) [ ]4
40
0
2 f x dx ln 2 x ln 24
= = . ln28= . .
20
( ) 3 2 2f x (1 )x x ( 1) x 2006,3
= + + \ . i) \ f 0x 1= ii) =1
) f(x)=0. ) fC , xx x=0 x=1.
ii) : f (1) 0. =
: ( ) 2 2f x (3 )x 2x ( 1) = + ( ) 2 2f 1 (3 ) 2 ( 1) 3 2 ( 1)( 2) = + = + = f (1) 0 = =1
=2.
-
2006
22
=1 ( ) 2f x 2x 2x = x - 0 1 + f + 0 - 0 + f . .
=2 ( ) 2 2f x x 2x 1 (x 1) = + =
x - 1 + f + 0 + f
=1.
iii) =1 ( ) 3 22f x x x 20063
= + . ) : f (( ,0]) ( , 2006] = 6017f ([0,1]) [ , 2006]
3=
[ 6017f ([1, )) , )3
+ = + f 0< . )
1 13 2
0 0
2 f(x)dx ( x x 2006)dx
3= = + =
4 3
10
x x 1 1 12035[ 2006x] 20066 3 6 3 6
+ = + =
21
f : \ \ : f
( )A 1,f(1) . f \ x 0
( ) xf (x)f x xex
= .
) ( ) 2x x xf x xe e e 12
= + . ) ( )2004f e f(2004). ) g \ ,
1
0
eg(0) g(f (x)) g(1 )2
.
-
2006
23
) f(0)=0 f (1) 0 = . *x \
x x x2
xf (x) f (x) f (x) f (x)e ( ) (e ) e c,x x x
= = = + x ( ,0) x (0, ) +
x1
x2
xe c x, x 0 f (x) , x 0, 0.
xe c x, x>0
+ +\
f 0x 0= x 0 x 0lim f (x) lim f (x) f (0) 0 + = = = 1 2 2f (1) 0 1 e c 1 0 c e = + = = f 0x 0=
x x1 2
x 0 x 0 x 0 x 0
xe c x 0 xe c x 0f (x) f (0) f (x) f (0)lim lim lim limx 0 x 0 x x 0 + +
+ + = = x x
1 2 1 2 1 2x 0 x 0lim (e c ) lim (e c ) 1 c 1 c c c + + = + + = + = .
1 2c c e= = =0, xf (x) xe ex = , x \ . x x xf(x)= (xe ex)dx xe dx exdx x(e ) dx e xdx = = =
2x x x x x x xxe (x) e dx e xdx xe e dx e xdx xe e e k
2= = = +
f(0)=0 0 1 0 k 0 k 1 + = = .
2x x xf (x) xe e e 1
2= + , x \ .
) 2004e 2004.> ( )( )20042004e 1 e 1 1 2004(e 1) 2004= + > + > Bernoulli. 1 : x \ ( )x 1f (x) ex e 1 . = f :
x - 0 1 + f (x) + 0 - 0 + f (x) 0 e1
2
H f [1, )+ , 20041 2004 e< < ( ) ( )2004f 2004 f e< .
2 : f ... 20042004,e , ( )2004 2004,e ,
20042004 2004
2004
f(e )-f(2004)f () f(e )-f(2004)=(e 2004)f ()e 2004
=
-
2006
24
N2004 2004 1
00 0
f(e )-f(2004)= (e 2004)e (e 1) 0>> >
> ( )2004 2004,e . ( ) ( )2004f e f 2004> . ) f [0,1] g \ , [ ]x 0,1
( ) ( ) ( )f . g .0 x 1 f(0) f(x) f(1) g f(0) g f(x) g f(1) ( ) ( ) eg 0 g f(x) g 1-
2 [ ]x 0,1 . g fD ,
( )1 1 10 0 0
eg(0)dx g f (x) dx g 1 dx2
( ) ( )1 1
0 0
e eg(0)(1 0) g f (x) dx g 1 (1 0) g(0) g f (x) dx g 12 2
.
22
) h \ h (x) 0 x \ .
) i) h 1-1 ii) h(x)=0 . ) h h(x) +h(-x)=0 x \ , \ , i) h(x)=0 .
ii)
0
h(x)dx 0= . ) f \ xf (x) xe x \ . )
x
0
g(x) f(x-t)dt= . )
x
0
g(x) f(x-t)dt= x x(x) xe e 1= + .
-
2006
25
) ) i) 1 2x , x \ 1 2x x 1 2x x< . h
[ ]1 2x , x , ... , ( )1 2 x , x , 2 1
2 1
h(x ) h(x )h ()x x
= .
h () 0 2 1 2 12 1
h(x ) h(x ) 0 h(x ) h(x )x x
. 1 2x , x \ 1 2x x 1 2h(x ) h(x ) , h 1-1. ii) h(x)=0 1 2 , 1 2 .
1 2h( ) h( ) 0= = . h 1-1 1 2 = . h(x)=0 .
) i) x=2
h h - 0 2h 0 h 02 2 2 2
+ = = = . x=2
h(x)=0 h 1-1.
ii) x \ h(x)=-h(-x). :
1 20 0
h(x)dx = h( x)dx= = (1), 20
h(-x)dx= . u=-x, du=-dx. x=0 u= x= u=0.
0
2 1 0
h(u)du = h(x)dx== (2). (1) (2)
1 1 1 10
2 0 0 h(x)dx 0= = = = . ) ) f(x-t) \ ,
x
0
g(x) f(x-t)dt= \ . u=x-t du=-dt. t=0 u=x t=x u=0.
0 x
x 0
g(x) f(u)du f(t)dt= = . H g \ g (x) f (x) = , x \ . ) gC , C
( ) y g(x) g(x) (x) 0 :y (x) y g(x)
= = = = . () , g(x) (x) 0 = .
xx x
0
g(x) (x) f(t)dt xe e 1 = + , x \ .
-
2006
26
h(x)= g(x) (x) , x \ , x
x x
0
h(x) f(t)dt xe e 1= + , x \ . x \ x x x xh (x) f(x) e xe e h (x) f(x) xe 0 = + = x \ , xf(x) xe x \ . h (x) 0 x \ ))ii) h(x)=0 .
00 0
0
h(0) f(t)dt 0e e 1 1 1 0= + = = . 0 h(x)=0 gC , C (0,g(0)) (0,0).
-
2006
27
23
f \ , 0 x\
( ) ( )x x x 21
xf x f t dt xe e x e 2 = + + + ) : i) f \ ( ) xx f x xe 2x = + ii) xf (x) e 2x 1= + + ) ( ) xg x xe 2x 1= + + , x\ , f g x 0= x 1= .
) i) f \ . ( )x1
f t dt \ ( ) ( )x
1
f t dt f x = .
( ) ( )x x x 21
xf x f t dt xe e x e 2 = + + + (1) ( ) ( )x ,0 0, + ( ) ( )x x x 2
1
1f x f t dt xe e x e 2x = + + + +
*\ f 0, \ .
(1) ( )( ) ( ) ( ) ( ) ( ) ( )x x x 21
x f x f t dt xe e x e 2 = + + +
( ) ( )x x x xf (x) xf x f (x) e xe e 2x xf x xe 2x + = + + = + . x\ , ( ) xx f x x e 2x = + . ) ii) ( ) xx f x xe 2x = + . ( ) ( )x ,0 0, + ( ) xf x e 2 = + ( ) ( )xf x e 2x = +
( ) ( )( )x
1
x2
e 2x c , x ,0f x
e 2x c , x 0,
+ + = + + +.
f \ , 0\ , : ( ) ( ) ( )
x 0 x 0lim f x lim f x f 0 + = = . ( ) ( )x 1 1
x 0 x 0lim f x lim e 2x c 1 c = + + = + ( ) ( )x 2 2x 0 x 0lim f x lim e 2x c 1 c+ + = + + = + .
( ) ( )1 2 1 21 c 1 c f 0 c c f 0 1+ = + = = = .
-
2006
28
( ) ( )xf x e 2x f 0 1= + + x\ . x 1= , ( ) ( )f 1 e 2 f 0 1= + + (1). ( ) ( )x x x 2
1
x f x f t dt xe e x e 2 = + + + . x 1= :
( ) ( ) ( )1 21
1 f 1 f t dt 1e e 1 e 2 1 f 1 e 3 = + + + = + (2). (1), (2): ( ) ( )e 3 e 2 f 0 1 f 0 2+ = + + = : ( ) xf x e 2x 1= + + . x\ . )
( ) ( )1 1 x x0 0
E f x g x dx e 2x 1 xe 2x 1 dx= = + + ( )1 1x x
0 0
E 1 x e dx e 1 x dx= = [ ]x 0,1 0 x 1 1 x 0
( ) ( ) ( )1 1x x0 0
E e 1 x dx e 1 x dx= = = ( ) ( ) ( )1 11x x 0 x
00 0
e 1 x e 1 x dx e 1 0 e dx = = + 1x 0
01 e 1 e e e 2 = + = + =
E e 2= . .
24
g :[,] R x (0, ) : g( x) g( x)+ = . : i. g() g()= ii. g
.
iii.
2
2
g(x)dx g(x)dx+
+=
-
2006
29
i.
x 0 w im g( x) im g(w) g()+ +
+ = =A A g 0x = , w=+x.
x 0 u img( x) img(u) g()
+ = =A A g 0x = , u=-x .
g() g()= . ii. . Rolle g [,].
(,) : g () 0 = . iii. u = x :
g( u) g(u)+ = , u (,) (i) g() g()= . :
g( x) g(x)+ = , x [,] . :
2 2
g(x)dx g( x)dx
+ += = +
y x= + , dy dx= x = y = x
2+= y
2+= .
:
2
2
I g(y)dy g(x)dx+
+= = .
25
f (x) nx x 1= +A , x > 0, > 0. i. f . ii. fC . ,
, . iii. , nx x 1 A .
i. :
1f (x) , x>0x
= 1f (x) 0 x
= =
-
2006
30
: x 0 1
+
f + 0 - f
ii. H fC 1 1 1M ,f , n
= A . , :
y n y nx1x
= ==
AA
iii. :
nx x 1, x 0 >A nx 1 , x 0
x+ >A
nx 1h(x)x+= A
. :
2 2
1 x ( nx 1) nxxh (x)x x
+ = =
A A
x 0 1 + h + 0 - h
h(1) 1= =
26
f : R R :
x f (t)0
f (x) 2 t e dt= , x R . i. f . ii. f 2f (x) n(x 1), x R= + A . iii. f ,
.
iv. : 2006x 0f (x)imx+
= +A .
-
2006
31
i. f f (t)h(t) te= t R , x f (t)
0F(x) te dt= .
x f (t)0
f (x) 2 te dt= . ii. , :
f (x)f (x) 2xe = f (x)e f (x) 2x = ( ) ( )f (x) 2e x = f (x) 2e x c= + 2f (x) n(x c)= +A (1)
x 0= : 0 f (t)0
f (0) 2 te dt 0= = (1) x = 0 :
f (0) nc= A c = 1 : 2f (x) n(x 1), x R= + A
iii.
22xf (x)
x 1 = + ,
x
f~
f
o
+
8 + 8
H f (0,f (0)) (0,0)= . . : ( )
xf ( ,0] [f (0), im f (x)) [0, ) = = +A ( )
xf [0, ) [f (0), im f (x)) [0, )++ = = +A f [0, )+ .
iv. L Hospital : 2
2006 2006x 0 x 0
f (x) n(x 1)im imx x+ +
+= =AA A
22
2005 2004x 0 x 0
2x1 x 1x 1im im
10032006x x+ + ++ = = +A A
-
2006
32
27
f : (0, ) R+
2x
x 1
f (t x)f (x) 1 dtx+= + , x>0.
: i. f . ii. f : f (x) ln x 1= + iii. f . iv. , , 0 < < < :
f () f () f () f () >
i. t x u = dt = du t = x + 1 u = 1 t = 2x u = x,
x x
1 1
f (u) 1f (x) 1 du f (x) 1 f (u)dux x
= + = +
x
1f (u)du f
. x
1
1f (x) 1 f (u)dux
= + (0, )+ . ii. : ( ) ( )x1xf (x) x f (u)du = +
1f (x) xf (x) 1 f (x) f (x)x
+ = + = f (x) nx c= +A f(1) = 1 c = 1. f f (x) nx 1, x 0.= + >A iii. 2
1 1f (x) , f (x) 0x x
= = < f . iv. ... f [,], [,] 1 1 (,), (,) 1 f () f()f ( )
= 2f () f()f ( )
= .
E 1 2 < f 1 2f ( ) f ( ) > f () f() f () f ()
> .
-
2006
33
28
f : (0, )+ \ : x>0, y>0 2 2f (xy) x f (y) y f (x)= + f 1 f(1) = 1. i. x > 0 :
h 1
f (xh) f (x) 2f (x)im xxh x x
= +A ii. f x > 0 :
2 3x f (x) 2xf (x) x = + iii. x > 0 :
2f (x) x ln x= iv.
x 0
f (x)imx
A v.
f, xx x = 1 x = e.
i. 2 2 2 2
h 1 h 1 h 1
f (xh) f (x) x f (h) h f (x) f (x) f (x)(h 1) x f (h)im im imxh x x(h 1) x(h 1) x(h 1)
+ = = + A A A
()
h 1
f (x)(h 1) f (h) 2f (x) 2f (x)im x x f (1) xx h 1 x x+ = + = + = + A
() ( x 1 x 1
f (x) f (1) f (x)f (1) im im 1x 1 x 1 = = = A A )
f(1) = 0 x = y = 1.
ii. 0x 0> 0
00 x x 0
f (x) f (x )f (x ) imx x = A ( 0x x h= )
0 0 00h 1 0 0 0
f (x h) f (x ) 2f (x )im xx h x x
= = +A ( 1 )
00 00
2f (x )f (x ) xx
= + 0x 0> f 22f (x)f (x) x xf (x) 2f (x) x
x = + = + 2 3x f (x) 2xf (x) x = + , x 0> .
iii. 2 3 2 3x f (x) 2xf (x) x x f (x) 2xf (x) x = + =
2
4x f (x) 2xf (x) 1
xx = 2
f (x) ( nx)x
= A , x > 0 2f (x) nx cx
= +A f(1) = 0 c = 0. 2f (x) x nx, x 0= >A .
-
2006
34
iv. ( )2x 0 x 0 x 0 x 0 x 0 x 0 x 0
2
1nxf (x) x nx nx xim im im(x nx) im im im im( x) 01 1x x 1
x xx
= = = = = = =
AA AA A A A A A A A
v. f, xx x = 1 x = e :
e e 21 1
E() f (x) dx x nx dx= = A nx 0A x [1,e]
2x nx 0A x [1,e] 3e e2
1 1
xE() x nxdx nxdx3
= = A A e3 3e
11
x xnx ( nx) dx3 3
= A A 3 3e
1
e 1 x 1ne n1 dx3 3 3 x
= =A A 3 e 2
1
e 1 x dx3 3
= = e3 3
1
e 1 x3 3 3
=
3 3e 1 e 13 3 3 3
= =
3 3 3e e 1 2e 13 9 9 9 9
= + = + .
29
f [,]
( )2 2 2 2
f (x)dx f (x) dx f () f ()+ = i. N f (x) f (x) 0 + = x [,] ii. ( ) ( ) 2 2 2 2 1f (x) dx f (x)dx f () f ()2 = = iii.
2 f e
2
++ = f.
-
2006
35
i. ( ) 22
I f (x)dx f (x) dx= + = ( ) 22
f (x)dx f (x) dx 2f (x)f (x)dx 2f (x)f (x)dx = + + =
( ) 22 2
f (x) f (x) 2f (x)f (x) dx f (x) dx = + + = ( ) ( ) 2 22 2 2
f (x) f (x) dx f (x) f (x) f (x) dx f () f () = + = + +
2 2 f () f ()= , ( ) 2
f (x) f (x) dx 0+ = . 0x [,] , 0 0f (x ) f (x ) 0+ ( )2f (x) f (x) 0+ x [,] , ( ) 2
f (x) f (x) dx 0+ > ,
. f (x) f (x) 0+ = x [,] .
ii. x [,] f (x) f (x) = , ( ) 2
f (x) dx f (x)f (x)dx f (x)f (x)dx = = =
( )2 2 2
f (x) 1 f () f ()2 2
= (1)
( ) 2 2 2 2 2 1f (x)dx f () f () f () f ()2+ = ( ) 2 2 2 1f (x)dx f () f ()2= (2)
(1) (2) ( ) ( ) 2 2 2 2 1f (x) dx f (x)dx f () f ()2 = = iii. x [, ]
xex xf (x) f (x) 0 f (x) e f (x) e 0
+ = + = ( )x x xe f (x) 0 e f (x) c f (x) ce = = = , x [,]
2 2 2 f e , c e e c e
2
+ + + ++ = = = x x f (x) e e f (x) e , x [,]+ + += = .
30
f [0,1] 1
0f (x)dx 0= ,
0x [0,1] 0f (x ) 0 . i. f(x) = 0 (0,1). ii. f [0,1] f(0) =
f(1) = 0, 1 2 , (0,1) 1 2f ( ) f ( ) 0 .
-
2006
36
i. f [0,1]. 0f (x ) 0> f (x) 0 x [0,1]> , 10 f (x)dx 0> .
0f (x ) 0< f (x) 0 x [0,1]< , o 10 f (x)dx 0 . f [0,1] f ^ [0,1] f 1-1 [0,1]. f (0) f () f (1) 0= = = Rolle [0,] [,1], 1 (0,) 1f ( ) 0 = 2 (,1) 2f ( ) 0 = . 1f ( ) =
'f 1 1
2f ( ) 1 = 2 , .
1 , 2 f (0,1),
1 2 1 2 , (0,1) f ( )f ( ) 0 .
31
[ ]f : , \ , , 0 , ( )f = ( )f x 0 < [ ]x , . ) f
-1f
) -1f ( )( )
( ) ( )f
-1
f
f t dt f t dt 0+ = i) ( )f ii) ( )0x , fC ( )( )0 0A x , f x ( )1 : x y 2006 0 + = ) i) ( ) , , ( )f = . ii) ( )1 2 , , ( ) ( )1 2f f 1 = .
-
2006
37
) ) ( )f x 0 < [ ]x , f [ ], 1-1, f . f -1f . f [ ], [ ], , [ ]( ) [ ] [ ]f , f (), f () f (),= = . [ ]1fA f (), = .
) i) ( )( )
( )f 1
f
I f t dt= . ( ) ( )1u=f t f u t = ( )f u du dt = . ( )t f = ( )11u = f f () = ( )t f = ( )12u = f f () = . ( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )f
-1
f
f t dt uf u du uf u u f u du f f f u du = = = ( )
( )
( ) ( ) ( )f
-1
f
f t dt f f x dx= (). ( )( )
( ) ( )f
-1
f
f t dt f t dt 0+ = () ( ) ( ) ( )
f f x dx f x dx 0 + = . ( ) f 0 = , 0 . ( )f = . ) ) ( )0x , ( )0f x 1 = ( ) ( )( )f x 1 f x x 0 = + =
(,). ( ) ( )g x f x x= + [,]. g [,] g (,) ( ) ( )g x f x 1 = + . ( ) ( )( ) ( ) ( ) ( )
g f g g
g f
= + = + == + = + .
Rolle ( )0x , ( ) ( )( ) ( )0 0 0 0g x 0 f x x 0 f x 1 = + = = .
) ) ( ) ( )h x f x x= [,]. h [,] . ( ) ( )h f = = ( ) ( ) ( )h f .= = = ( ) ( ) ( )2h h 0= < .
Bolzano ( ) ( )h x 0 f x x 0= = (,), ( ) , ( ) ( )f 0 f = = . ( ) ( )h x f x 1 = ( )f x 0 < ( )h x 0 < . h [,]. ( ) , .
-
2006
38
) f ... [,] [,]. ( )1 , ( )2 , : ( ) ( ) ( ) i)1
f f f = = (1) ( )
( ) ( ) i)2
i)
f f f = = = (2)
(1), (2) ( ) ( )1 2 f f 1 = = .
32
( ) 2f x x 1= + , x\ . ) : i) ( )x
xlimf x+
ii) ( )x 0xlim
f x 1 B) ) f .
) ( )1 2+1
0
xI dxf x
= , 0,1,2,3,...= i) 0 1 , ii) ( ) 12 1 2 2 + = , 2,3,...= 3 .
) i) +
2 2
x 1x 1 x 1
+ + . 2 2 21 x 1
x 1 x 1 x 1 + + + .
2x
1lim 0x 1+
=+ ( )xxlim 0f x+
= . ii) 0
( ) ( )( )( )( )2 2
22 2 2
x x 1 1 x x 1 1xg xxx 1 1 x 1 1 x 1 1
+ + + += = =+ + + +
( ) 2x x 1 1g xx x
+ += . ( )2x 0
x 1 1lim
x
+ += ( )2
x 0
x 1 1lim
x+
+ += +
( ) 2x 0 x 0
x x 1 1lim g x limx x
+ += =
( ) 2x 0 x 0
x x 1 1lim g x limx x+ +
+ += = +
2x 0
xlimx 1 1 + .
) ) ( ) ( )22 2 2
x 1 2x xf x2 x 1 2 x 1 x 1
+ = = =+ + + .
-
2006
39
( )2
xf x 0 0 x 0x 1
= = =+ . f ( ],0 f [ )0,+ . f ( )f 0 1= . ) i)
1 12
0 2 00
xI dx x 1 2 1x 1
= = + = + . 0 2 1= 1 13
21 2 2
0 0
x x dx x dx
x 1 x 1= = =+ +
( ) ( ) ( )1 112 2 2 2 2 200 0
x x 1 dx x x 1 x x 1dx = + = + +
( ) ( ) ( )13
2 21 12 2 2
00
x 12 x 1 x 1 dx 2 3
2
+ = + + =
32 2 4 2 2 22 2 2 23 3 3 3 3
= + = + = . 1 2 2I 3=
ii) 1 12+1
2 2 2
0 0
x xI dx x dxx 1 x 1
= =+ + ( ) ( )1 1 112 2 2 2 2 2 2-1 2
00 0 0
I x x 1 dx x x 1 x x 1dx 2 2x x 1dx = + = + + = + =
( )1 1 1 1 2 1 12 2 1 2 1 2 1
2-1
2 2 2 20 0 0 0
x 1 x x x x2 2 x dx 2 2 dx 2 2 dx 2 dxx 1 x 1 x 1 x 1
++ ++ += = = + + + + 1I 2 2I 2I = 1I 2I 2 2I + = ( ) 12 1 I 2 2I + = , 2,3,...= 3= 3 27I 2 6I= (2) 2= 2 05I 2 4= . 0 2 1= ( )25I 2 4 2 1= 2 25I 2 4 2 4 5I 4 3 2= + = . 2 4 3 2I 5= (2) (1), (2)
( )3
6 4 3 27 2
5
= 335 5 2 24 18 2= +
335 23 2 24= . 3 23 2 24 35= .
-
2006
40
33
) i) f, g [,] ( ) ( )f x g x [,]
: ( ) ( )
f x dx g x dx ii) :
1
t0
1 dt 12e 1 2e 1 3
+ + ) f [ )0,+ ( ) ( )f 0 f 0 1= = x 0 : ( )( ) ( ) ( ) ( ) ( )2f x f x f x f x f x + =
i) ( )( )2 xf x 2e 1, x 0= . ii) : ( )( )
x
20
dt2x 12 f t
=+ (0,1).
) i) f, g [,]. g f [,] ( ) ( )g x f x 0 [,]. ( ) ( )( ) ( ) ( ) ( ) ( )
g x f x dx 0 g x dx f x dx 0 f x dx g x dx ii) ( ) t1f t 2e 1= + , t\ . H f \ ( ) ( )
t
2t
2ef t 02e 1
= .
f \ [0,1]. ( ) ( ) ( )0 t 1 f 0 f t f 1 ( )0 11 1f t 2e 1 2e 1 + + t
1 1 12e 1 2e 1 3
+ + .
) i) 1 1 1
t0 0 0
1 1 1dt dt dt2e 1 2e 1 3
+ + 1 1 1
t0 0 0
1 dt 1dt dt2e 1 2e 1 3
+ + [ ] [ ]1
1 1t0 0
0
1 dt 1t t2e 1 2e 1 3
+ + 1
t0
1 dt 12e 1 2e 1 3
+ + . ) i) x 0 ( )( ) ( ) ( ) ( ) ( )2f x f x f x f x f x + =
( )( ) ( ) ( ) ( )( ) ( ) ( )f x f x f x f x f x f x + = ( ) ( )( ) ( ) ( )f x f x f x f x = . ( ) ( ) ( )h x f x f x= , x\ . ( ) ( )h x h x = x\ . ( ) xh x c e= , c\ ( ) ( ) xf x f x ce = . x 0= ( ) ( ) 0f 0 f 0 ce 1 1 c 1 c 1 = = = .
-
2006
41
( ) ( ) xf x f x e = x\ . ( )( ) ( )2
xf x e2
= .
( )( )2 x1
f xe c
2= + , 1c \ .
x 0= , ( )( )2 0 1 1 1f 0 1 1e c 1 c c2 2 2= + = + = . ( )( )2 xf x 1e2 2
= ( )( )2 xf x 2e 1, x 0= .
) ii) ( ) ( )( )x
20
dt x 2x 1, x [0,1]
2 f t= + .
( )( )2 tf t 2e 1= . ( ) x t0
dt x 2x 1
2e 1= + .
[0,1] ,
( ) ( )0 t0
dt 0 2 0 1 0 1 0
2e 1= = ( ) ( ) 0 1 0< . . Bolzano
( ) ( )( ) ( )( )x x
2 20 0
dt dt x 0 2x 1 0 2x 1
2t f t 2t f t= = =
(0,1).
( ) ( ) x t0
dt x 2x
2e 1
= + ( ) x1
x 22e 1
= + , ( )x
x
4e 1 x 0
2e 1+ = >+
x\ . \ , [0,1]. ( ) x 0= (0,1).
( )( )x
20
dt2x 12 f t
=+ (0,1) .
-
1-16 2008.doc
-1-
2008
1 z^ , ,\ , ( ) i1 iz
i++ = + (1)
) z . ) z
. ) z .
) 4 z 3 4i 7< + <
) 1 2z ,z ^ (1) 1 2z z 2 ) z\ , (1)
( ) i i1 iz 1 iz i i
+++ = + =+ + 21 iz 1 1 iz 1 1 z 1 z 0 + = + = + = = ,
z 0= (1) ( ) ( ) i 1 i 0 i+ = + = =+ , .
) (1)
21 iz 1 iz i 1 i z i 1 z i 1+ = = = = ,
z
( )K 0,1 1= .
) z 2= , z 2i= z 0= , z 0=
2
y
K
x
) ( )z 3 4i z 3 4i + =
(z) z ( )A 3, 4 . ,
, :
( ) ( ) ( )AB AM , ( ) 2 2 3 5 34= + =
( ) ( ) 34 1= = , ( ) ( ) 34 1= + = + ,
34 1 z 3 4i 34 1 + + , 34 1 7+ < 4 34 1< ,
4 z 3 4i 7< + <
2
y
K
x
y
x
()
A(3, 4)
) , 1z , 2z z i 1 = , ( )1 2z z 2 2 = =
-
1-16 2008.doc
-2-
2 1z , 2z , 3z 1 2 3z z z = = =
31 2
2 3 1
zz z 1Re Re Rez z z 2
= = =
: ) 1 2 3z z z 0+ + = ) 1z , 2z , 3z .
) 1 1 12 2 2
z z z2Rez z z
= + 21 2 2 1
1 2 1 2 22 2
z z z z12 z z z z z2 z z
+ = + = 21 2 1 2z z z z + = (1). 22 3 2 3z z z z + = (2) 21 3 1 3z z z z + = (3) 1 2 3 1 2 3z z z 0 z z z 0+ + = + + =
( )( )1 2 3 1 2 3z z z z z z 0 + + + + =
( )( )1 2 3 1 2 3z z z z z z 0 + + + + = 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3z z z z z z z z z z z z z z z z z z 0 + + + + + + + + = ( ) ( ) ( )2 2 21 1 2 2 1 2 2 3 3 2 3 1 3 3 1z z z z z z z z z z z z z z z 0 + + + + + + + + = 2 2 2 2 2 2 0 + + = , . 1 2 3z + z + z = 0. ) 1 2 2 3 3 1z z z z z z . = = 2 1 3z z z= :
1 2 2 3z z z z = 1 1 3 1 3 3z z z z z z+ + =
( )( ) ( )( )2 21 3 1 3 1 3 1 3 1 3 1 32z z z 2z 2z z 2z z z 2z z 2z + = + + + = + +
1 1 1 3 1 3 3 3 1 1 1 3 3 1 3 34z z 2z z 2z z z z z z 2z z 2z z 4z z + + + = + + +
2 21 2 1 23 z 3 z z z = = , . 1 2 2 3z z z z . =
2 3 3 1z z z z = , 1 2 2 3 3 1z z z z z z . = = 1z , 2z , 3z .
-
1-16 2008.doc
-3-
3
2z z 0 + = , z^ , ,\ 1z , 2z , 1z 2 i= +
1) , .
2) 2008 20081 2z z+ .
3) ( )1A z , ( )2B z , ( )3 z 1z , 2z , 3z ( )13
2
z 1z 17 iz 5
= + + , : ) .
) 1 1w - z = w - z , w\ .
) w,
2 2w - z + w - z = 10 , .
1) 2z z 0 + = (1) , , .\
1z 2 i= + (1) 2 1z z 2 i= = , Vieta : 1 2
z z (2 i) (2 i) 41+ = + + = =
2 21 2z z (2 i)(2 i) 2 1 51
= + = + = =
2) ( )20082008 2008 20081 2 1 1u z z z z= + = +
( )20082008 2008 20081 2 1 1u z z z z= + = + ( )2008 20081 1z z= + u=
( ) ( )u u u u 0 2i Im u 0 Im u 0 u= = = = \
3) ) ( ) ( )132
z 1 2 i 1z 17 i 17 iz 5 2 i 5
+= + + = + +( ) ( ) ( )
2
2 2
2 i 1 4 4i 1 117 i 17 i2 1 5 5 5+ + = + + = + + =+
3 4i 17 i 20 5i 4 i5 5
+ + + += = = +
1z , 2z , 3z ( )A 2,1 , ( )B 2, 1 ( ) 4,1 . : ( ) 2 1AB z z 2 i 2 i 2i 2 i 2= = = = =
( ) 2 23 2 z z 4 i 2 i 2 2i 2 2 8 2 2= = + + = + = + = =
( ) 1 3 z z 2 i 4 i 2 2= = + = = AB = . 2 2 2 4= = , 2 2 2 4= = , 2 8= ,
2 2 2 = + , .
-
1-16 2008.doc
-4-
) 1 1w z w z = 1 2 1 2w z w z w z w z = = z z
1 2w z w z= = (1)
( )
: w x yi= + , x, y\ , ( )M x, y (1) :
x yi 2 i x yi 2 i+ = + + ( ) ( ) ( ) ( )x 2 y 1 i x 2 y 1 i + = + +
( ) ( ) ( ) ( )2 2 2 2x 2 y 1 x 2 y 1 + = + + ( )2x 2 2y+ 2y 1 + ( )2x 2= 2y+ 2y 1+ +
y 0 = . w\ . ) 2 2w z w z 10 + = 2 1w z w z 10 + =
z z
2 1w z w z 10=
+ =
( )2 1w z w z 10 2 5 2 + = = = (1)
2 1z z 2 1 2 = =
2 2< (1) . 4
f :\ \ ( )f 0 = 1 z^ .
( ) -2008 -2006 2007 -2009f x i + 2xi = 2z + 2 i - z -1 i ,x\ :
) ( ) 2f x = x + 1 , x\ z -1 = 2z + 2
) z . ) f
( )O 0, 0 , ( )A , 0 , ( )2 , + 1 ( )2 0, + 1 , > 0.
f . ) ( ) 2008 2006 2007 2009f x i 2xi 2z 2 i z 1 i + = + (1)
( )2008
5022008 4 502 4
1 1 1 1i 1i i 1i
= = = = = , 2006 2006 4 501 2 21 1 1i 1i i i
+= = = =
2007 4 501 3 3 2i i i i i i += = = = , 2009 2009 4 502 1 21 1 1 ii ii i i i
+= = = = =
( ) ( )1 f x 2x 2z 2 i z 1 i = + + ( )f x 2z 2 i 2x z 1 i + + = + ( )f x 2x
2z 2 z 1
= + =
x\ ( ) ( ) ( )2f x 2x f x x = = , c\ ,
( ) 2f x x c= + , x\ .
x 0= ( ) 2f 0 1 0 c 1 c 1= + = = , ( ) 2f x x 1= + , x .\
-
1-16 2008.doc
-5-
) ( )2z 2 z 1 2 z 1 z 1 2 z 1 z 1+ = + = + = (2)
z x yi= + , x, y\ ( )M x, y , (2) : 2 x yi 1 x yi 1 2 x 1 yi x 1 yi+ + = + + + = +
( ) ( ) ( )2 22 2 2 2 2 22 x 1 y x 1 y 4 x 2x 1 y x 2x 1 y+ + = + + + + = + +
2 2 2 24x 8x 4 4y x 2x 1 y 0 + + + + = 2 2 2 2 103x 3y 10x 3 0 x y x 1 03
+ + + = + + + = (3)
(3) 2 2x y Ax By 0+ + + + = 103
= , 0= 1.=
2
2 2 10 100 64 4 4 1 4 03 9 9
+ = = = > .
(3) ,2 2
5 , 03
2 2 4 4
2 3+ = = . z
5 ,03
43
= )
y
(0, +1)2
x
y
x
(, +1)2
(, 0)
1
Cf
fC x x
x 0= x = , :
( ) ( ) ( )2 x 1 02 20 0 0
E f x dx x 1 dx x 1 dx+ >= = + = +
E ( )2E 1 = = + :
( ) ( ) ( ) 2 20
1 1E E x 1 dx 12 2
= + = + ( )3
2
0
x 1x 13 2
+ = +
( ) 02 21 1 1 13 2> + = +
2 21 1 1 1 3 2 2
+ = + 2 3= 3=
-
1-16 2008.doc
-6-
5 f \ ( )f 1 >0 ( ) ( )
( )f 1 f 0 4 3
z = + i2 f 1
, ( )zz = 1 3i .2
) ( ) ( ) ( )f 1 f 0 f 1 = 8
) 2Re ( )z = z
) ( ) ( )f 1 < 2 < f 1
) f ( )11 < f z < 0 ) 1 < 0 < 1 f \ , ( ) ( ) ( )f 1 < f 0 < f 1 ,
( )f 1 >0 , ( ) ( ) f 1 > f 0 0> (1).
( ) ( ) ( ) ( ) ( )f 1 f 0z 1 4 3z = 1 3i + i 1 3i =2 2 2 f 1 =
( ) ( ) ( ) ( )( ) ( )f 1 f 0 3f 1 f 06 2 3
= + + i4 f 1 f 1 4
z ,
( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 f 1 f 02 3 0 8 3 3f 1 f 0 f 1 = 0 f 1 f 0 f 1 = 8
f 1 4 = (2)
) ( ) ( )
( )f 1 f 0 6
z = +4 f 1
( ) ( ) ( )
8f 1 =
f 1 f 0
(2), ( ) ( ) ( ) ( ) ( ) ( ) ( )f 1 f 0 6f 1 f 0z = + = f 1 f 0 = 2Re z
4 8.
) (1) (2) , ( ) ( ) ( )f 1 , f 0 , f 1 2 8. 8. 2
( )f 1 . 2 8. , 2 ( )f 1 . ( ) ( )f 1 < 2 < f 1 .
-
1-16 2008.doc
-7-
) f , 1 1 , .
( ) ( ) ( ) ( ) ( ) ( ) ( )11 < f z < 0 f 1 < z < f 0 f 1 < f 1 f 0 < f 0
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )
22 2
2
f 1 < f 1 f 0 f 1 < f 0f 1 < f 1 f 0 < f 0
f 1 < f 0f 1 f 0 < f 0
( ) ( )f 0 f 1 > 0.> . 6o
) w^ , w w 0+ + = , ,,\ . : i ) w w 0+ + = i i ) w\ ) z 3 32z z 5zz 7 0,+ + = :
) :
i ) 3 3z z zz 1= = i i ) z 1= ) z .
) i ) w w 0+ + = (1) (1) w w 0+ + = (2) i i ) (1) (2)
(w w) (w w) 0 = ( )(w w) 0 w w = =
( ) ( )w w 0 2i Im w 0 Im w 0 w \ = = = ) ) i ) 3 32z z 5zz 7 0+ + = (3)
(3) 3 32z z 5zz 7 0+ + = (4) (3) (4)
3 3 3 33zz 3z z 0 zz z z = = (5)
(3) (5) 3 3zz z z 1= =
i i ) 3z z 1,= 3z z 1= z 0
3 3 4z z 1 z z 1 z 1 z 1
= = = =
) 1z 1 zz
= = 3z z 1= 3 21z 1 z 1 z iz= = =
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1-16 2008.doc
-8-
7
f \ g :
( ) ( )( )( )7 5
6 4
1 - f x + x + 1g x =
f x + 4x + 2, \
) : ( )
xlim g x ( )xlim g x+
) ( )xlim g x+
) f(x) 0 x\ ( )f 2008 < 0, ( )xlim g x = +
) ( ) [ ]f 0 , 1 ( ) ( )h x = g x + 2008 + , x\ , \
( )xlim h x = 1+ , ( )f .
) ( ) ( )( )( )7 5
6 4
1 f x x 1g x
f x 4x 2 + += + + , \
( )xlim g x ( )xlim g x+ g
( ), ( ),+ . g ( ) ( )( )P x
g xQ x
=
( ){ }g 0 0D x : Q x 0= =\ \ g ( ), ( ), .+ ) ( )f 0 ( )f 1 :
( ) ( )( )( )( )
( )7 5
6 4x x x
1 f x x 1 1 f lim g x lim lim x
f x 4x 2 f + + + + + = = = + + +
x
xlim)(f
)(f1
xlim x .+ = +
:
i) ( )( ) ( )( ) ( ) ( )1 f
0 1 f f 0 0 f 1f > > < < ( )
xlim g x+ = +
ii) ( )( ) ( ) ( )1 f
0 f 0 f 1f < < > ( )
xlim g x+ =
-
1-16 2008.doc
-9-
iii) ( )f 1= ( ) 56 4x 1g x x 4x 2+= + +
( ) 5 56 4 6x x x xx 1 x 1lim g x lim lim lim 0x 4x 2 x x+ + + ++= = = =+ +
iv) ( )f 0= ( ) 7 54x x 1g x 4x 2
+ += + , ( )7
34x x x
x 1lim g x lim lim x4x 4+ + +
= = = +
( )( )
( ) ( )( )x
, 0 f 1
lim g x , f 0 f 1
0, f 1+
+ =
) ( )f x 0 x\ f \ , \ , ( )f 2008 0< ( )f x 0< x\ ( )f 0< .
( ) ( )( )x x1 f
lim g x lim xf =
xlim x = , ( )f 0< , ( )1 f 0 > , ( )( )
1 f 0
f < ( )
xlim g x = +
) ( ) ( ) ( ) ( )h x g x 2008 h x g x 2008= + + = + () ( ) [ )f 0,1 ( )
xlim g x+ = +
( )xlim h x+ = + , ( )xlim h x 1+ =
( )f 1= ( )( ) ( )( )
x xlim h x lim g x 2008+ + = + ( )xlim g x 2008+= + 0 2008= + 2008= ,
( )( )
xlim h x 2008+ = ( )x xlim h x lim 2008 1 2008+ + = = 2007 =
8 ( ) 12 xxf x = x + - 2x , > 1 .
) : ( )
xlim f x , ( )-x 0lim f x , ( )+x 0lim f x , ( )xlim f x+
) f . ) f.
) \ ( )f x = 0.
-
1-16 2008.doc
-10-
f ( ) ( )A ,0 0, .= +
) 1
2 xxlim x
= + , 2
xlim x = +
x
1u1 x ux1x u 0lim 0x
lim lim
=
==
x 0
0 1= = .
xxlim 0 = ( )
1
xlim 2x
>
= + , ( )xlim f x = +
x 0
1u1 x2 ux
21 ux 0 limx
1lim x lim 0u
=
=
= = , 2u
1lim 0u
= uulim 0 = .
x 0x 0lim 1 = = ( )x 0lim 2x 0 = , ( )x 0lim f x 1 = .
x 0
1u1 ux2 x
21 ux 0 limx
lim x limu+
+
=
+ =+
= = ( )( )
u u
D.L. u u2
lnlim lim2uu
+ + + +
= = =( )2u
D.L.H u
lnlim
2
+ +
+= = + .
xx 0lim 1+ = ( )x 0lim 2x 0+ = , ( )x 0lim f x+ = + .
( ) 1 xxx x
lim f x lim x x 2x+ +
= + = +
) x A f
( ) 1 12 xx x 1f x 2x x ln ln 2x
= + + = 1 1
xx x2x ln ln 2 +
f ( ) ( ) ( )11
21 xx 2xx2
ln lnf x 2 2 lnx x
= + + =
( )21 2xx ln ln 2 2 lnx x
= + + ( )21 2xx ln 1 1 ln
x = + +
( )f x 0> x A , f ( ),0 ( )0,+ . ( )f 1 0. = x 1> , f ( )0,+ , ( ) ( )f x f 1 > , ( )f x 0 > . 0 x 1< < ( ) ( )f x f 1 < ( )f x 0 < . ( )
x 0lim f x ln 2 0
= < , 1>
ln 1
ln 2 ln 2 2 ln 2 0 . 1< < <
-
1-16 2008.doc
-11-
f ( ),0 ,
( )( ) ( ) ( )x x 0f ,0 lim f (x) , lim f (x) , ln 2 = = . ( )f x 0 < ( )x ,0 ,
ln 2 0 . <
f :
x
f( )x
f( )x
+0
1
1
f ( ),0 , ( ]0,1
[ )1, .+ f 0x 1= ( )f 1 0= .
) ( ) ( ) [ ) [ )f A 1, 0, 0,= + + = + .
) ( )f x 0 x 1= = 9
f :\ \ ( )f =\ \ : H f .\ H f + ( ) ( )f xf x + e = x .x\
:
) ( )xlim f x+ = +
) H f .\ ) H f -1f
) H f .\ ) fC , x x x = 1 x = e + 1 3E = .
2 .
) ( )xlim f x+ = ( ) ( )( )f xxlim f x e+ + = , xlim x+ = + , .
( )
xlim f x+ = A \ ( ) ( )( )f xxlim f x e e+ + = + AA , xlim x+ = + , .
f + ( )xlim f x .+ = +
) x\ , ( ) ( )( ) ( )f xf x e x + = , ( ) ( )( ) ( ) ( )f x f x1f x 1 e 1 f x 01 e + = = >+ , f .
-
1-16 2008.doc
-12-
) f 1 1 , :
( ) 1y f x x f (y)= = , y 1y e f (y)+ = , ( )1 xf x e x , x R. = + ) f \ ( )f x
11 e+ \ , ( )f x
\ , f \ ( ) ( ) ( )( )( )f x
2f x
f x ef x 0
1 e
= ), f \ .
) f x 1 ( ) ( )f x f 1 0, = ( ) ( )e 1 e 11 1
E f x dx f x dx+ +
= = ( ) ( )1u f x x f u= = , ( )( ) ( ) ( )1 u udx f u du e u du e 1 du = = + = +
x 1= ( )u f 1 0= = , ( ) ( )1f 0 1 f 1 0 = =
x e 1= + ( )u f e 1 1= + = , ( ) ( )1f 1 e 1 f e 1 1 = + + = ,
( ) ( ) 11 1 1 1 1 1 21u u u u u0
0 0 0 0 0 0 0
u 3E u e 1 du ue du udu u e du udu ue e du .2 2
= + = + = + = + = 10 f :\ \ :
( )( )
2
x f x + 5 1=1 + f x 2
x\ ( )f 0 = 3 . ( ) 2f x = x + x + 9 , .x\
. ( ) ( )g x = lnf x :
) g.
) g 4 20
1I = dxx + 9
) J + 9I = K , :
4 220
xJ = dxx + 9
4 20
K = x + 9 dx
) J + K = 20
) J, K.
) g -1g .
. x\ :
( )( )2x f x 5 11 f x 2 + = + ( ) ( ) ( ) ( )2 21 f x 2x f x 10 f x 2x f x 9+ = + =
( ) ( ) ( )( ) ( )22 2 2 2 2 2f x 2x f x x x 9 f x x x 9 h x x 9 + = + = + = + (1),
( ) ( )h x f x x= , x\
-
1-16 2008.doc
-13-
x\ ( ) ( )2 2x 9 0 h x 0 h x 0+ > > h , \ .
x 0= ( ) ( )h 0 f 0 0 3 0= = > , ( )h x 0> , x\ ,
( ) ( ) 21 h x x 9 = + ( ) ( )2 2f x x x 9 f x x x 9 = + = + + , x\
. ) x\ 2 2x x 9 x x x x 0+ + > + = + . A = \ ) x\ ( ) ( ) ( )22 2 21 1 2xg x x x 9 g x 1x x 9 x x 9 2 x 9 = + + = + + + + + + ( )
2
1g xx x 9
=+ +
2x x 9+ + ( )2 2
1g xx 9 x 9
=+ +
( ) ( )4 4 402
0 0
1 9I dx g x dx g x g(4) g(0) ln9 ln3 ln ln 33x 9
= = = = = = = + )
4 4 4 42 22
2 2 20 0 0 0
x 1 x 9J 9I dx 9 dx dx x 9dx Kx 9 x 9 x 9
++ = + = = + =+ + +
) ( ) ( )4 442 2 200 0
K x x 9 dx x x 9 x x 9 dx = + = + + =
( )4 4 222 2
0 0
1 x20 x x 9 dx 20 dx 20 J2 x 9 x 9
= + = = + + . J K 20+ =
) : J K 9IJ K 20 = + =
9J 10 ln 3J K 9ln3 2J K 20 9K 10 ln 3
2
= = + = = +
) x\ ( )2
1g x 0x 9
= >+ . g ,
1 1 , . ( )y g x= x.
( ) ( )ye x 22 2 y 2 y 2 yy ln x x 9 x x 9 e x 9 e x x 9 e x>= + + + + = + = + =
( )2y 2y2 2y y 2 y 2y 1y ye 9 e 9x 9 e 2xe x 2xe e 9 x f y2e 2e + = + = = =
2y
y y 2y 2y 2yy
e 9e x e 2e e 9 e 92e> > > > , y\
1f : \ \ ( ) 2x1 xe 9f x .2e=
-
1-16 2008.doc
-14-
11
z = x + i x\ , 2 1w = zz
f \ \ .
) 0x , 2 1z z .
) < < , ( ) 0f = x , ( )f = 1 ( )f > 0 , \ , ,
( )f = 0 ) : ( )1
0
Re w dx
) ( )22 2 22 2 21 1 x i x 1w z x i x 1 2xi x 1 2x iz x i x 1 x 1 x 1 = = + = + = + + + + + + , x .\
2 1zz
3212x 0 2x 2x 1 0.x 1+ = + + =+
( ) 3g x 2x 2x 1= + + , x\ :
g [ ]1, 0
( ) ( ) ( )g 1 g 0 3 1 3 0. = = <
Bolzano, ( )g x 0= ( )ox 1,0 .
x\ ( ) 2g x 6x 2 0 = + > , g \ , ox . 0x , 2 1z z .
) f \ , [ ] , . .. ,
( ) , , ( ) ( ) ( ) 0 0f f 1 x x 1f + = = =
0< > , ( )0 0 0x 1,0 1 x 0 x 1 0 < < + > , ( )f 0 < .
:
f [ ] ,
( ) ( )f f 0 <
Bolzano, \ , ( )f 0. =
) ( ) [ ] ( )11 1 1 1 1 13 12 2 22 2 200 0 0 0 0 00
x x x 1 1Re w dx x 1 dx x dx dx dx x x 1 dxx 1 x 1 3 2 x 1
= = = + = + + +
( ) 120
2 1 2 1 4 3ln 2 4 ln8ln x 1 ln 23 2 3 2 6 6
+ + = + = = = .
( )10
4 ln8Re w dx6+= .
-
1-16 2008.doc
-15-
12
f :\ \ ( )x 1
f 2x + 1 7lim = 10.
x 1
1) : ) ( )f 3 = 7 ) ( )f 3 = 5
2) () f ( )( )M 3,f 3 .
) () y = 5x 8. ) , 3, ().
2 m sec ,
.
1. ) ( ) ( )f 2x 1 7h xx 1+ = ( ) ( ) ( )f 2x 1 x 1 h x 7+ = +
( ) ( ) ( )x 1 x 1limf 2x 1 lim x 1 h x 7 7 + = + = (1)
2x 1 + = , ( ) x 1 3 , ( ) ( )f 3
(1) limf 7 f 3 7. = =
) x 1 ( ) ( )f (3) 7x 1 x 1
f 2x +1 7 f 2x +1 f (3)lim = 10 lim = 10
x 1 x 1
=
(2)
2x 1 + = , ( ) x 1 3 , (2) 3
( ) ( ) ( ) ( ) ( ) ( ) 3 3 3
f f 3 f f 3 f f 3lim 10 lim 10 lim 2 10 1 3 31
2 2
= = = ,
( ) ( ) 3
f f 3lim 5
3 = , ( )f 3 5. =
2. ) f ( )( )M 3,f 3
( ) ( ) ( )( ) ( ) : y f 3 f 3 x 3 y 7 5 x 3 y 5x 8 = = =
) ( ) x, y (). :
( ) ( ) 3 71 1OM det OM, | |x y2 2= = =JJJJG JJJG ( )1 13y 7x 3 5x 8 7x2 2 = =
( )1 8 x 3 4 x 3 4x 122
= = = , ( )x 3> :
( )( )( ) ( )( ) ( ) 2E x t 4x t 12 4x t 4 2 8 m sec = = = =
-
1-16 2008.doc
-16-
13 ( ) xf x = x + e 1
1) f .
2) xe 1 x= 3) g :\ \ x\
( ) ( )g xg x + e = 2x + 1.
) g .
) ( )g 0 0=
4) ( )( )g f x 0>D
5) f 1fC ( )M e,1 .
1fC .
1) x\ ( ) xf x 1 e 0 = + > , f \ . 2) ( )x xe 1 x e x 1 0 f x 0= + = = . x 0.= f , 11 .
3) ) g , 1 2 gx ,x A 1 2x x< ,
( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( )1 21 2
1 2 g x g x1 2 1 2 1 2g x g x
g x g xg x e g x e 2x 1 2x 1 x x
e e
+ + + + + , g
.
) ( ) ( )g 0g 0 e 1+ = ( )g 0 0= g 11 . 4) ( )( ) ( )( ) ( ) gg f x 0 g f x g 0 > > ( ) ( ) ( ) ff x 0 f x f 0 x 0> > >
5) f 11 . 1fM C , , y x= ,
( ) fN 1,e C . fC : ( ) : ( ) ( )( ) ( )y f 1 f 1 x 1 y 1 e x 1 = = + () y x= 1fC
1.
1 1y x1 e 1 e
= ++ +
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1-16 2008.doc
-17-
14
( ) ( )2x 3
22
ln t 1f x dt
t 1
= + 1) f.
2) f .
3) ( )x 5
f xlim 0
x 5=
4) fC x x , .
5) ( )z f 3 2i= . z .
1) ( ) ( )2ln t 1g t t 1= + ( ) ( )
x
2
F x g t dt .=
g ( )gA 1, ,= + ( )2 1, + , F ( )FA 1, .= +
( ) 2h x x 3= , x ,\
( ) ( )( ) ( )( ) ( )2x 3
22
ln t 1f x F h x F h x dt
t 1D
= = = +
( ){ } ( ) ( ){ } { }2 2f h FA x A / h x A x / x 3 1, x / x 3 1\ \= = + = > = { } { }2x / x 4 x / x 2\ \= > = > = { }x / x 2 x 2 > < \
( ) ( )fA , 2 2, .= + 2) f ( ) 2h x x 3=
( ) ( )x2
F x g t dt= g. ( ) ( )x , 2 2, +
( ) ( ) ( )( ) ( )( )
( )2 2 2x 3
22 22 2 2
2
ln x 4 2x ln x 4ln t 1f x dt x 3
t 1 x 3 1 x 3 1
= = = + + +
f
2 2f
x = 0 A x = 0 x = 0f (x) = 0
x 4 = 1 x = 5 x = 5 A
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1-16 2008.doc
-18-
f (x) f
.
f
f
+5 2
2 5x
f ( , 5 (2 , 5 ,
)5 , 2 )5 , . + f
1x 5= 2x 5= ( ) ( )f 5 f 5 0. = = 3) x 5 :
( ) ( ) ( )( )0 20
2D.L.H 2x 5 x 5 x 5
2x ln x 4f x f xlim lim lim 0
1x 5 x 3 1 = = = +
(1 )
( ) ( ) ( ) ( ) ( )( )2
22x 5 x 5
2 5 ln 5 4f x f 5f xlim lim f 5 0
x 5 x 5 5 3 1
= = = = + (2 )
4) ( )( )
f 5 0
f 5 0
= =
( )( )
f 5 0
f 5 0
= = fC x x .
5) ( )3 5, + f ( ) ( )f 3 f 5 ,> ( )f 3 0.> ( )z = f 3 2i , 4 . 15
[ )f : 0,+ \ : ( ) ( )x
1
f tf x 1 1 dt
t = + + x 0> (1)
1) :
) ( )f x xln x x= + , x 0>
) ( )f 0 0=
2) f .
3) fC .
4) ( )e
1
f x dx1 2e
e 1
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1-16 2008.doc
-19-
1) ) f ( )0 , + , ( )x1
f t1 dt
t +
( )f t1t
+ ( )0 , .+ ( )x 0 , +
( ) ( ) ( ) ( ) ( ) ( )x1
f tt
f xf x = 1+ 1+ dt f x = 1+ xf x = x + f x
x
( ) ( ) ( ) ( ) ( ) ( )2xf x f x f x f x1 ln x ln x c f x x ln x cxx x x x = = = + = +
(1) x 1= ( )f 1 1,= ( )f 1 ln1 c c 1.= + = ( )f x x ln x x ,= + x 0>
) f 0
( ) ( ) ( ) ( )D.L.Hx 0 x 0 x 0 x 0 x 0
2
1ln x 1 xf 0 lim f x lim x ln x x lim lim lim x 0.1 1
x x
+ + + + +
+
+= = + = = = =
( )f 0 0=
2) ( )x 0 , + ( )f x ln x 2 = + . ( ) 2 21f x 0 ln x 2 0 ln x 2 x e x e = + = = = = ( ) 2 21f x 0 ln x 2 0 ln x 2 x e x e > + > > > >
f
x
f
f
01
e2
..
f 210 ,e
, 21 ,e + ,
1x 0= ( ) ( )f 0 0= 2 21x e= 221f ee
=
3) ( )x 0 , + 1f (x)x
= f 0
f [ )0 , .+
( )( )1 1A x ,f x , ( )( )2 2B x ,f x , ( )( )3 3 x ,f x ,
( ) ( ) ( ) ( )2 1 3 2
2 1 3 2
f x f x f x f x
x x x x = = ( )
f ... [ ]1 2x , x [ ]2 3x , x , ( )1 1 2 x , x , ( ) ( ) ( )2 11
2 1
f x f xf
x x =
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1-16 2008.doc
-20-
( )2 2 3 x , x , ( ) ( ) ( )3 223 2
f x f xf
x x = .
( ) ( ) ( )1 2f f = , f (x) 0, > f
, " 1 1 ".
4) f [ ]1, e , 1 x e ( ) ( ) ( )f 1 f x f e ,
( ) ( ) ( ) ( )e e e e e e1 1 1 1 1 1
f 1 dx f x dx f e dx 1dx f x dx 2edx
( ) ( )( )ee
1
1
f x dxe 1 f x dx 2e e 1 1 2e
e 1
16
f : \ \ ( ) ( )x 2
f tx
tf x dt1 e
= + x\ . 1) :
) f ( )40160
I f x 2008 dx= ) f . ) ( ) 3xf x , x
3= \ .
2) z iwz 1+= + , z^ z 1
w , : ) z y x 1= .
) ( )( )M ,f fC 1 0 < < z.
1) ) fA = \ , x\ ( ) ( ) ( ) ( )
x x2 2
f t f tx x
t tf x dt dt f x1 e 1 e
= = = + + f .
( )40160
I f x 2008 dx= . u x 2008= , du dx.=
x 0= 1u 2008= x 4016= 2u 2008= ,
( ) ( ) ( ) ( )2008 2008f u f u u xdu dx
2008 2008
I f u du f u du= =
= = = =
x 2008= 1u 2008= x 2008= 2u 2008= ,
( )( ) ( ) ( )2008 2008 20082008 2008 2008
f u du f u du f u du I
= = = =
2I 0 I 0= =
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1-16 2008.doc
-21-
) ( ) ( ) ( ) ( ) ( )c x x x2 2 2 2
f t f t f t f tx c c c
t t t tf x dt dt dt dt1 e 1 e 1 e 1 e
= + = ++ + + + , c\ x .\
f , ( ) ( )x 2
f tc
t x dt1 e
= +
( ) ( )2
f t
tg t1 e
= + \ , .
( ) ( )x 2
f tc
tK x dt1 e
= + ,\
( )h x x= ( ) x . ) x\
( ) ( )( ) ( ) ( )2 2 f
f x f x
x xf x x1 e 1 e = + =+ + ( ) ( )
2 2
f x f x
x x1 e 1 e
+ =+ +
( )( )
( )( ) ( )
( )( )( )f x2f x2 2 2 2
2f x f x f x f x
f x
x 1 ex x x e x x .1 1 e 1 e 1 e 1 e1e
+= + = + = =+ + + ++
( ) 3xf x c3
= + , x\ . x 0= ( )f 0 0= , c 0= . ( ) 3xf x
3= , x .\
2) ) ( )( ) ( )( )z i z iw w w z i z 1 z 1 z iz 1 z 1+ = = + + = + + +\
( ) ( )z z z iz i z z iz z i z z i z z 2i 0 + + + = + + + + = , z x yi= + 2yi 2xi 2i 0 y x 1.+ + = = z y x 1.= ) ( ) ( )f x x 1 f x x 1 0= + + = ( )1,0 . ( ) ( )h x f x x 1= + + , [ ]x 1,0 .
h [ ]1,0 .
( ) ( ) 1h 1 f 1 03
= = < , ( ) ( )h 0 f 0 1 1 0= + = > ( ) ( )h 1 h 0 0 <
Bolzano, ( ) 1,0 , ( )h 0= .
( ) 2 21x x 2lim e lim ee+ + + = =
-
17-32 2008.doc
-22-
2008
17
f \ ( ) ( )f 0,= +\ ,
( ) ( )f x f x 1 = x\ .
) :
) f .\
) f .\
) ( ) ( )g x f x= .\
) ( )( )( ) ( )2 21
0
f x f 0 f 1dx
f x 2 =
) ( )f 0 1= f .
) ) x\ ( ) ( )f x f x 1 = (1)
f ( )0,+ , x\ ( )f x 0 > , (1)
( )f x 0 > , f .\ ) 1 2x ,x \ ( ) ( ) ( ) ( ) ( ) ( )
f :
1 2 1 2 1 2 1 21 2
1 1x x x x f x f x f x f xf x f x
< > > < >
( ) ( )g x f x= .\
) x u x u , = = dx du.= x 0= u 0= x 1= u 1.=
( )( )( )( )
1 1
0 0
f x f udx du
f x f u
= = ( ) ( ) ( ) ( ) ( )12 2 21
0 0
f u f 0 f 1f u f u du
2 2
= = ) (1) x ,\ x ,\ :
( ) ( ) ( ) ( )f x f x 1 f x f x 1 = = ( ) ( )f x f x 1 = (2)
(1) (2) :
( ) ( ) ( ) ( ) ( ) ( )f x f x f x f x 0 f x f x 0 + = = ( ) ( ) 1f x f x c = , x\
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17-32 2008.doc
-23-
x 0= ( ) ( ) 1 1f 0 f 0 c c 1 = = . ( ) ( )f x f x 1 = , x\ (3)
(1) (3) x\ ( ) ( ) ( ) ( ) ( ) ( ) ( )f x 0f x f x f x f x f x f x > = =
( ) ( ) ( ) ( )xe x xf x f x 0 f x e e f x 0 = = ( )( ) ( )x x 2f x e 0 f x e c = =
x 0= ( ) 0 2 2f 0 e c c 1. = = ( ) ( )x xf x e 1 f x e = = , x .\ 18
) xe x 1 0 + > x .\
) f \ ( )f 0 2,= x\
( ) ( )( ) ( ) ( ) ( )xf x f x e x 1 f x f x =
) ( ) xf x e x 1= + , x .\
) e e e e < ,` .<
) , ,
( ) ( )x 2 A f x B f x = + x .\
)
( ) x x 2g x e x 1= + , x x x 0= x 1.=
) ( ) xh x e x 1= + , x .\ x\ ( ) xh x e 1 =
( ) x x x 0h x 0 e 1 0 e 1 e e x 0 = = = = =
( ) x x x 0h x 0 e 1 0 e 1 e e x 0 > > > > > h
x
h x( )
h(x)
0
2
( )h 0 2=
h 0x 0= ( )h 0 0= , x\ :
( ) ( ) ( )h x h 0 h x 2 , ( ) xh x 0 e x 1 0> + >
) ) x\ ( ) ( )( ) ( ) ( ) ( )xf x f x e x 1 f x f x =
( ) ( ) ( ) ( ) ( )x xf x e f x e x f x f x f x 0 + + = ( ) ( ) ( ) ( )x xf x e x 1 f x e 1 0 + = ( ) ( ) ( ) ( )x xf x e x 1 f x e x 1 0 + + = ( ) ( ) ( ) ( )( )
x x
2x
f x e x 1 f x e x 10
e x 1
+ + = +
( ) ( )x xf x f x0 ce x 1 e x 1 = = + + ( ) ( )xf x c e x 1 .= +
x 0= ( ) ( )0f 0 c e 0 1 c 1= + = . ( ) xf x e x 1= + , x .\
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17-32 2008.doc
-24-
) ( ) xf x e x 1= + [ )0,+ ( () ),
,` < ( ) ( )f f < e 1 e 1 e e + < + <
e e e e e e e
e
e e e ee e e ee e ee
< < < < ) , \ , x\ ( ) ( )x 2 A f x B f x = + ( ) ( )x xx 2 A e 1 B e x 1 = + + ( ) xx 2 A B e Bx B A = + +
x\ , x.
x 0= ( ) 00 2 A B e B 0 B A 2 A = + + = B B A+ + 2B 2 B 1 = =
x 1= ( ) 11 2 A B e B 1 B A = + + ( )1 A B e B = + B+ B 1A =
( ) ( )( )1 A 1 e A A 1 e 1 0 A 1 = = = ) g [ ]0,1 ( )g x 0< [ ]x 0,1 , gC , x x x 0= x 1= :
( ) ( )1 1 x0 0
x 2E g x dx dxe x 1
= = = + ( ) ( )x x1
x0
e 1 e x 1dx
e x 1 + = +
( )x1 1 1 1x x
x x x0 0 0 0
e x 1e 1 e x 1dx dx dx 1dxe x 1 e x 1 e x 1
+ += + = + = + + + ( ) ( ) ( )1x
0ln e x 1 1 1 0 ln e ln 2 1 = + + = + = ln 2 ..
19
( ) 1xf x e= , x 0.>
i) ( )x 0lim f x+
ii) g ( ) 1x1g x 1 ex
= + , x 0> ( )g 0 0.=
) g [ ]0 , 1 .
) ( )10
I g x dx.=
i) f ( )0 , + ( ) 1 1 1x x x21 1f x e e ex x = = =
-
17-32 2008.doc
-25-
( ) ( )x 0
1u1 2x2 ux
2 u1 u u D.L.Hx 0 x 0 limx
1 ulim f x lim e lim u e limx e+ +
+
+ = + + + =+
= = = =
( )( )
( )( )
2
u uu u D.L.H u uu u
u 2u2u 2lim lim lim lim 0e ee e
+ + + + + +
= = = = =
ii) ) g [ ]0,1 ( )0,1
( ) ( )x 0lim g x g 0+ = ( ) ( )x 1lim g x g 1 =
g ( )0,1 . :
( ) ( )1 1x 1x 1 x 1
1 1lim g x lim 1 e 1 e g 1x 1
= + = + = , g 0x 1.=
( ) ( )1 2x 1 1 1D.L.Hx 0 x 0 x 0 x 0 x 0 x 01x x xx 2
11 1111 1xx xlim g x lim 1 e lim lim lim lim 0 g 0 ,x 1e e ee x
+ + + + + +
+ +
++ = + = = = = =
x 0
1u1 xux
1 ux 0 limx
lim e lim e++
=
+ =+= = + 1x 0
x
1lim 0,e
+= g 0x 0.=
f [ ]0 ,1 . ) 1
( ) 1x1g x 1 ex
= + [ ]0,1 [ ]0,1 :
( )1xxe , 0 x 1G x
c, x 0
< = = ( ) 1x
x 0 x 0c lim G x lim xe 0,+ +
= = = c 0=
( ) ( ) ( ) ( ) ( )1 1 10
0 0
I g x dx G x dx G x G 1 G 0= = = = = 1 10e e = 2
( )1 1 1x0 0
1I g x dx 1 e dxx
= = + 1 1
t
x 0x
1lim 1 e dtt+
= + 1 11 1
t t
x 0x x
1lim e dt e dtt+
= + =
( )1 11 1t tx 0
x x
1lim t e dt e dtt+
= + 1 1 11 1 1
t t t
x 0x xx
1lim t e t e dt e dtt+
= + =
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17-32 2008.doc
-26-
1
1 x
x 0lim e xe t+
= 2
1t
1 1t
x
e dt 1 1t
x
1 e dtt
+ 1x
x 0
1 1lim xee e+
= =
1x
1 1x 0 x 0 x 0x x
x 1lim xe lim lim x 0e e
+ + +
= = = ,
x 0lim x 0+ = 1x 0
x
1lim 0.e
+=
20 ( ) x 2
0
dtf xt 1
= +
) f .
) ( ) 2xx f x x 1< < + x 0.< ) ( )f x x= x ,
2 2 ( ).f 1
) fC y 0= , x=0 x=1.
) ( ) 21g t t 1= + ,\ 0 ,\ x .\
fA .\=
x\ ( ) x 2 20
dt 1f x 0,t 1 x 1
= = > + + f .\
( ) ( ) ( )fx 0 f x f 0 f x 0< < < , ( ) ( ) ( )fx 0 f x f 0 f x 0.> > >
x\ ( ) 2 21 2xf x x 1 x 1 = = + +
( ) 22xf x 0 0 x 0x 1 = = =+ ( ) 22xf x 0 0 x 0x 1 > >
-
17-32 2008.doc
-27-
( ) x,0 ( ) ( ) ( ) ( ) ( )f 0 f x f xf f 0 x x = =
f ( ],0 , f ( ],0 ,
( ) ( ) ( ) ( ) ( )x 02 2f x f x1 xx 0 f x f f 0 1 xx 1 x x x 1 , ( ) ( ) ( ) ( ) ( ) ( )1 1 1 11
00 0 0 0
E f x dx 1 f x dx x f x dx x f x x f x dx = = = = =
( ) ( ) ( )( )1 1 12 22 20 0 0
1 1 1 1f 1 x dx x 1 dx ln x 1 dxx 1 4 2 x 1 4 2
= = + = + =+ +
( ) 120
1 1 2ln 2 ln 4ln x 1 ln 2 .4 2 4 2 4 4
= + = = = 21
( ) ( )2x 1
20
1F x dt4 t 1
=
+
) F.
) i) ( ) F x x6
= x ,2 2
ii) fC : ( ) ( )21f t
4 t 1=
+,
x x x=0 x 3 1.=
) 3
21
dxI4 x
= .
-
17-32 2008.doc
-28-
) ( ) ( )21f t
4 t 1=
+ ( )3 ,1 , ( )0 3 ,1 , F
x 3 2x 1 1 2 2x 2 1 x 1 < < < < < < , ( )FA = 1 , 1 .
( )x 1 , 1 ( ) ( ) ( )2 2 21 2 1F x 2x 1
4 4x 1 x4 2x 1 1 = = = +
) i) x ,2 2
:
( ) ( )2 2
1 1F x x x 11 x x
= = = ,
( ) ( ) ( )F x x F x x c = = + (1)
( )2x 10
f t dt x c
= + , x ,2 2
x6
= , ( )00
f t dt c c6 6
= + = , (1) ( ) F x x
6= , x ,
2 2 (2)
ii) ( )f t 0> , t 0, 3 1 ( ) ( )2 1
3 1 3
0 0
E f t dt f t dt
= = =
( ) ( ) ( )2 1
3 1 3 2
0 0
E f t dt f t dt F 3 3 6 6
= = = = = .
) (ii) ( )3 1
20
1 dt64 t 1
=
+ (3) x t 1= + , dx dt .= t 0= x 1= t 3 1= x 3.= (3)
3
21
1 dx64 x
=
22 f \ , : ( )( ) ( )2 2f x x f x x 3 0 + = x\
) x = f , ( )f 2=
.
) f .
) fC + .
) H f \ x = f , ( )f 0 = (1)
x\ ( )( ) ( )2 2f x xf x x 3 0 + = (2)
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(2) ( ) ( ) ( )2f x x f x 2x f x 0 + = (3)
x = (3) ( ) ( ) ( ) ( ) ( )f 02f f 2 f 0 f 2. = + = = (4)
x = (2) ( )( ) ( ) ( )f 22 2 2 2 2 2 2f f 3 0 4 2 3 0 3 3 1 1.= + = + = = = = ) fC ( )( )0 0M x ,f x , ( )0f x 0 = (5) (3) ( )( ) ( ) ( ) ( )22 f x 2f x 2 2f x x f x 0 + + = (6)
0x x= (6) ( )( ) ( ) ( ) ( )20 0 0 0 02 f x 2f x 2 2f x x f x 0 + + = (5) ( )( ) ( )20 0f x f x 1 0 + = , ( )0f x ,
3 0,= < f . ) fC + ( )x f xlim x+ = \ ( )xlim f x x ,+ = \ x 0 ( )( ) ( )2 2f x x f x x 3 0 + = ( ) ( )
2
2
f x f x 31 0x x x
+ = , :
( ) ( )2
2x
f x f x 3lim 1 0x x x+
+ = 2 1 0 + = , fC + .
23
( )f , g : 1, + \ ( ) ( )f 0 g 0 1= = , ( )x 1 , + ( ) ( ) ( ) ( ) ( ) ( )2 22f x f x g x 2g x g x f x 0 + = + = (1) , ( )f x 0
( )g x 0. ) ( ) ( )f x g x 0= > ( )x 1 , . + ) ( ) 1f x =
x + 1.
) f .
) E()
f , x x x = , x 1= + 0> , ( )
lim E .+
) (1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )g x 02 2 22f x f x g x 0 2f x g x f x g x 0 + = + = (2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )f x 02 2 22g x g x f x 0 2g x f x f x g x 0 + = + = (3)
(2), (3) ( ) ( ) ( ) ( )2 22f x g x f x g x + ( ) ( ) ( ) ( )2 22g x f x f x g x= + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )g x 0f x g x g x f x f x g x g x f x 0 = =
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17-32 2008.doc
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( ) ( ) ( ) ( )( )( )( )
( )( )2
f x g x g x f x f x f x0 0 c
g x g x g x
= = = , ( )x 1 , + .
x 0= ( )( )f 0 1c c 1g 0 1
= = = , ( )( ) ( ) ( )f x
1 f x g xg x
= = , ( )x 1 , . + f , g ( )1, + , ( ) ( )f x 0 g x 0 ( )x 1, . + f , g
( ) ( )f 0 g 0 1 0= = > ( ) ( )f x 0 g x 0.> > ) ( ) ( ) ( ) ( ) ( ) ( ) ( )f x g x2 32f x f x g x 0 2f x f x 0= + = + = ( ) ( ) ( )f x 032f x f x =
( )( ) ( ) ( ) ( )( ) ( ) ( )3 23 22f x 11 2f x f x 1 f x x x cf x f x = = = = +
x 0= ( )21 0 c c 1
f 0= + =
( )x 1 , + ( ) ( )f (x) 02 1 1f x f xx 1 x 1
>= =+ + ) ( )x 1 , + ( ) ( )( )
( )( )( ) ( )( )2x 1 x 11 1f x 0
x 1 2 x 1 x 1 2 x 1 x 1x 1
+ + = = = = x 1 0+ >
x 1lim x 1 0+ + = . x 1= fC .
( )x 0 , + ( )x x x
1f x 1 1x 1lim lim lim 0
x x xx 1+ + + += = = = + \
( )( ) ( )x x x
1lim f x x lim f x lim 0 x 1+ + +
= = = = + \
y 0x 0 0= + = fC + .
) ( ) ( ) ( ) ( ) 1 1 1f x 0 1
1E f x dx f x dx dx 2 x 1 2 2 2 1x 1
+ + +> + = = = = + = + + + ( ) ( )( )
2 2 1 2 1lim lim 2 2 1 lim
2 1+ + ++ + + + + = + + = = + + +
( ) ( ) ( )2 2
2 2 1 2 2 1 2lim lim lim 0. 2 1 2 1 2 1+ + +
+ + + = = = =+ + + + + + + + + 24
*f : \ \ ( )f x 0 *x\ 1-1 : ( ) ( )1
1f xf x
= x 0 f ( ) 0,= + , :
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17-32 2008.doc
-31-
) ( )( ) 1f f x x= ( ) 1f x f 1x = x 0 ) ( )f 1 1= ( )f 1 1 = ) ( )1f x x = . ) f , :
i ) ( )f x 0< x 0> ( )f x 0> x 0< ii ) H f ( ),0 .
) ( ) ( )11f x
f x = x 0 (1)
(1) x ( )f x 0 :
( )( ) ( )( ) ( )( ) ( )( )11 1 1f f x x f f x
xf f x f f x = = = , x 0 (2)
(2) x 1 0x :
1f f xx
= , ( ) ( )( )f x 0
1 1 11 1f f f f x f f xx x
= =
( ) ( ) ( ) ( ) ( )111 1f x f f x f x f x f 1x x
= = x 0 (3) ) (3) x 1= ( ) ( ) ( ) ( ) ( )2f 1 f 1 1 f 1 1 f 1 1 f 1 1= = = =
( )f 1 1= , ( ) ( ) ( )1 11f 1 1 f 1 1
f 1 = = = , f
( )0,+ , ( ) ( ) ( ) ( ) ( )11x 1 f x f 1 f x 1 1 f x 1
f x> > > < < ,
( )1f 1 1 = . ( )f 1 1= (3) x 1= ( )( ) ( ) ( )( ) ( ) ( ) f :1 1
f f 1 1 f f 1 f 1 f 1 1
= = = ) x 0 :
( ) ( ) ( ) ( )1 11 1 1f x x f
