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1
2 2001
:
: (4)
1o
A.1. z1, z2 . : z1 z2 = z1 z2.
7,5
.2. , .
z :
. z z z 2 =
. zz 22 = . z - z = . z z = . z z i =
5
.1. i, 3 - 1 z i 4 3 z 21 =+= , .
1
2
1. z z 21 . 4 2. z 21 . 2
3. 2
2z . 25
4. 1z . 5 5. z i 2 . 2
. 5
. 10 7,5
.2. 1, z z =
z
1 z = .
5
2 f :
3 x ,
3x
e-1
3 x ,x f(x) 3-x
2
>
=
. f , = 1/9. 9
. Cf f (4, f(4)).
7
2
3
. f, xx x=1 x=2.
9 3 f, R, :
f3(x) + f2(x) + f(x) = x3 2x2 + 6x 1 x R,
, 2 < 3 . . f .
10 . f .
8 . f(x) = 0
(0,1). 7
4 f, R, o :
i) f(x) 0, x R ii) f(x) = , x R. dt(xt)ftx 2 - 1 1
0
22 g
x - f(x)
1 g(x) 2= , x R.
3
4
. (x)2xf - )x(f 2= 10
. g . 4
. f :
x1
1 f(x)
2+= . 4
. (x f(x) 2x). lim x +
7
( ) 1.
( , , ). . , .
2. . .
, .
3. . 4. . 5. : (3)
. 6. : (1)
.
K
4
1
30 2002
: : (4)
1o
A. f ' [, ]. G f [, ],
.
)(G)(Gdt )t(f =
12
.1. f(x) = x. f R
f(x) = x .
8
.2. , . . f [,]
(,], f [,] .
1
. , 1-1 , .
1
1
2
. f x0 , 0f(x)
x xlim
0
= . 0 f(x)
x xlim
0
= 1
. f R ,
. dx)x(xf)x(xfdx)x(f = 1
. f(x) > 0 x , 0 f(x)x x
lim
0
> 0 . 1
2 z f() = i z, IN*.
. f(3) + f(8) + f(13) + f(18) = 0 . 7
. z= Arg(z) = ,
f(13) =
++
+ 2
i2
.
8
. z= 2 Arg(z) = 3
,
0, z f(13).
10
2
3
3 f, g R . fog 1-1.
. g 1-1. 7
. : g(f(x) + x3 - x) = g(f(x) + 2x -1) .
18
4 . h, g [, ]. h(x) > g(x) x [, ],
.
dx)x(gdx)x(h >
2 . R f,
:
x R f(0) = 0 . ,1xe)x(f )x(f =
) f f. 5
) , f(x)x f(x) 2
x 0.
12
) f, x = 0, x = 1 xx,
)1(f 2
1 E
4
1
4
( )
1. ( , , ). . , .
2. . .
, .
3. . 4. . 5. : (3)
. 6. : (1 1/2)
.
K
4
1
29 2003 :
:
1o
A. , f x0, .
8
.
;
7
. , .
. z _z ,
zzz == . 2
. f .
f(x)>0 x , f .
2
1
2
. f, ,
, c IR . c)x(fdx)x(f += 2
. f , f .
2
. f x0 . f x0 f(x0)=0, f x0.
2
2 z=+i, ,IR w=3z
_zi +4,
_z z .
. Re(w)=3+4
m(w)=3. 6
. , w y=x12, z y=x2 .
9
2
3
. z, y=x2, .
10
3 f(x) = x5+x3+x .
. f f .
6
. f(ex)f(1+x) xIR. 6
. f (0,0) f f 1.
5
. f 1, x x=3 .
8
4 f [,] (,). f() = f() = 0 (,), (,), f()f()
4
. f(x)=0 (,).
8
. 1, 2 (,) f(1)0.
9
. f .
8
( )
1. ( , , ). .
2. .
.
. 3. . 4.
. 5. : (3)
. 6. : 10.30 .
K
4
1
8 2003 :
: (4)
1o
A. f . F f , :
.
G(x) = c)x(F + , c R f
. G f
G(x) = c)x(F + , c R . 10
. , . . z1, z2 ,
z z z z z z 212121 ++ . 2
. f ' (, ), x0, f .
f (x) > 0 (, x0) f (x) < 0 (x0, ), f (x0) f .
2
1
2
. f : R 11 , x1, x2 A :
x1 = x2, f(x1) = f(x2) .
2
. f, g , :
= dx g(x) (x) f g(x) f(x) dx (x) g f(x) . 2
. x = x0 f ;
7
2 . ()
z :
2 z = m (z) 0 . 12
. , z (),
z
4 z
2
1 w
+=
xx . 13
2
3
3
x 1 x f(x) 2 += . . 0 f(x)lim
x=+ .
5
. f, x .
6
. 0 f(x) 1 x (x) f 2 =++ . 6
. ( ) 1 2 ln dx 1x
1
1
0 2 +=+ . 8
4 f IR , :
f(x) = )x2(f f (x) 0 x IR . . f .
8
. f(x) = 0 .
8
. (x) f
f(x) g(x) = .
g xx, 45 .
9
3
4
( )
1. (, , ). .
2. .
.
.
3. .
4. .
5. : (3) .
6. : 10.00 .
K
4
1
1
27 2004 :
: TE (4)
1o
A. f ' x0 . f x0 , f(x0)=0
10
. f x0 =
5
. .
. .
2
. ` ? f(x)lim0xx , ` ?? -/ f(x)limf(x)lim
00xxxx
2
. f, g x0, fg x0 :
(fg)(x0) = f(x0) g(x0) 2
2
2
. f, . f(x)@0 x , f .
2
. f [,]. G f [,],
/? )(G)(Gdt)t(f 2
2
f f(x)=x2 lnx .
. f, .
10
. f .
8
. f. 7
3 g(x)=exf(x), f
IR f(0)=f(23 )=0 .
. (0,23 )
f()=/f(). 8
3
3
. f(x)=2x2/3x, I()= dx)x(g
0
, IR 8
. )(I lim -
9
4 f: IR IR f(1)=1. x IR ,
g(x)= )1x(z
1z3dt)t(fz
3x
1 /-/ 0,
z=+iC, , IR *, :. g
IR g.
5
. N z
1zz -?
8
.
Re(z2) =2
1/ 6
. A f(2)=>0, f(3)= >, x0 (2,3) f(x0)=0.
6
4
4
( )
1. (, , ). . .
2. , . .
, .
3. .
4. .
5. : (3) .
6. : 10:30 .
K
1
1
5 2004 :
: (4)
1o
A. f .
f f(x) = 0 x , f .
9
. , .. f x0
, .
2
. .
2
. f, g IR fog gof, .
2
2
2
. C C f f1 y = x xOy xOy.
2
. f x0,
f(x) lim )x(flim k0x x
k
0x x = , f(x) 0 x0, k k 2.
2
. f (, ) [, ].
6
2 f: IR IR f(x) = 2x + mx 4x 5x, m IR , m > 0.. m f(x) 0 x IR .
13
. m = 10, f, xx x = 0 x = 1.
12
3 f: [, ] IR [, ] f(x) 0 x [, ] z Re(z) 0, m(z) 0 Re(z) >Im(z) .
3
3
z
1 z + = f()
z
1 z
22 + = f2(), :
. z= 1 11
. f2() < f2()
5
. x3f() + f() = 0 (1, 1).
9
4 f [0, +) IR ,
+= 210 2 dt 2xf(2xt) 2x f(x) .. f (0, +).
7
. f(x) = ex (x + 1). 7
. f(x) [0, +). 5
. f(x)lim x + f(x)lim x .
6
4
4
( )
1. (, , ). .
2. , . .
, .
3. .
4. .
5. : (3) .
6. : 10:00.
K
1
1
31 2005 :
: (4)
1o
A.1 f, [, ].
f [, ] f() f() f() f() , x0 (, ) ,
f(x0) = .
9
.2 y = x + f +;
4
B. , . . f [, ] f() < 0
(, ) f() = 0, f() > 0.
2
. ( )g(x)f(x)lim0
xx+ ,
f(x)lim x x 0
g(x)lim x x 0
.
2
2
2
. f f1 f y = x, f1 .
2
. f(x) > 0 x0 f(x)lim0x x
= 0, += f(x)
1 lim
x x 0
.
2
. f ,
( ) ) f(- f(x) dt)t(fx
= x . 2
. f , x x , .
2
2
z1, z2, z3 z1=z2=z3= 3. . :
z
9 z
11 = .
7
. z
z
z
z
1
2
2
1 + . 9
. : z1 + z2 + z3= 3
1 z1 z2 + z2 z3 + z3 z1.
9
3
3
3
f f(x) = ex, > 0.
. f .
3
. f, , y = ex.
.
7
. () , f, yy,
() = 2
2 - e .
8
. 2
() lim
2
+
+ .
7
4
f IR ,
2 f(x) = ex f(x) x IR f(0) = 0. . :
2
e 1 ln f(x)
x
+= .
6
. N : x
dt t) - f(x lim
x
0
0 x
.
6
4
4
. :
h(x) = dt)t(f t x
x 2005 g(x) =
2007
x
2007
.
h(x) = g(x) x IR . 7
. 2008
1 dt)t(f t
x
x 2005 =
(0 , 1).
6
( )
1. ( , , ). .
2. , . .
, .
3. . 4. . 5. : (3)
. 6. : 10:30 .
K
1
1
6 2005
:
: (4)
1o
A.1 f x)x(f = . f (0,+) :
x2
1)x(f = .
9
.2 f:A IR 1-1; 4
B. , . . ,
f 0, f .
2
. f (,) xo. f (,xo) (xo,) , ( ) )x(f,x oo f.
2
2
2
. .
2
. f,g fog gof, fog gof.
2
. xx.
z,z 2
. f IR *, :
= dx)x(fdx)x(f . 2
2
. z1, z2
z1+z2=4+4i , i55zz2 21 += z1 , z2 .
10
. A z,w z 1 3i 2 w 3 i 2 : i.
z, w , z=w 10
ii. z w. 5
3
3
3
f, IR f(x)0 x IR . . f 1-1.
7
. Cf f (1,2005) (-2,1),
( ) 2)8x(f2004f 21 =+ . 9
. C f, Cf
(): 2005x668
1y += .
9
4
f: IR IR , 2005
x
x)x(flim
20x= .
. :
i. f(0)=0
4 ii. f(0)=1.
4
4
4
. IR , : ( )( ) .3)x(fx2 )x(fxlim 2222
0x=+
+
7
. f IR f(x)>f(x) x IR , :
i. xf(x)>0 x0. 6
ii. .
1
27 2006
:
: (4)
1o
A.1 f, .
:
f(x)>0 x , f .
f(x) 0)x(f >
x0.
2
1
2
. H f() f .
2
. , x IR . 1 -xx 3 x)3( = 2
.
f (x)g(x)dx=[f(x)g(x)] (x)g(x)dx, f,g f [,].
2 2
f(x) =2+(x-2)2 x2.
. f 1-1.
6
. f -1 f .
8
. i. f f -1 y=x.
4
ii. f f-1.
7
2
3
3
1 z,z,z 321321 zzz === . z z 0z 321 =++
. : i. 321321 zzzzzz == .
9
ii. 4zz2
21 Re . 1)zz( 21 8
. z1 ,z2,z3 , .
8 4
f(x)=1x
1x
+ lnx.
. f.
8
. N f(x)=0 2 .
5 .
g(x)=lnx (,ln) >0 h(x)=ex (,e) IR , f(x)=0.
9 .
g h .
3
3
4
( )
1. ( , , ). . .
2. , . .
.
3. . 4.
. 5. : (3)
. 6. : 10.30 .
K
4
1
5 2006
:
: (4)
1o
A.1 : (x)=x, xIR . 10
.2 f . f ;
5
B. , . . z1, z2 , :
2121 z z z z + . 2
. f, g xo
g(xo)0, g
f
xo :
[ ] 2)g(x )g(x )(x f)(x g )f(x )(x gf o ooooo =
. 2
. x0 [ ]x
1 xn =l .
2
1
2
. f: IR 11, y f(x)=y x .
2
. f [,]. G f [,], . G() G() f(t)dt =
2
2
1x
x
e1
e 1 f(x) ++
+= , xIR . . f
IR . 9
. dxf(x)
1 . 9
. x
3
. () f x=2 yy yo=3,
i. ().
9
ii. f, (), xx
5
3x = .
9
4
f(x) = nx1)(x1)n(xx ll ++ x>0. . i. : 0x ,
x
1 nx 1)n(x >
4
( )
1. ( , , ). . .
2. , . .
.
3. . 4. . 5. : (3)
. 6. : 10.30 .
K
4
1
24 2007 :
: (5)
1o
A.1 z1 , z2 , :
2121 z zz z = . 8
.2 f, g ;
4
.3 y = f +; `
3
B. , , , , , . . f [,]
x[ , ] f(x) 0 . > 0 dx f(x) 2
. f x . f f(x) > 0 x .
2
1
2
. f x0 g x0 , gof x0 .
2
. f , ( ) ( ) (x) g g(x)f dt f(t) g(x) = .
2
. > 1 . 0 lim x x
= 2
2
2ii 2
z ++= IR .
. z (0,0) =1.
9
. z1, z2
2ii 2
z ++=
= 0 = 2 .
i. z1 z2 .
8
2
3
ii. :
)z( )(z 22
1 = .
8
3
:
f(x) = x3 3x 22
IR + 2
, Z .
. f , .
7
. f(x) = 0 .
8
. x1 , x2 x3 f, (x1 , f(x1)) , B(x2 , f(x2)) (x3, f(x3)) y = 2x 22 .
3
. f y = 2x 22 .
7
3
4
4
f
[0, 1] f(0) > 0.
g [0, 1]
g(x) > 0 x [0, 1]. :
F(x) = , x [0, 1], x0 dt g(t) f(t) G(x) = , x [0, 1]. x0 dt g(t)
. F(x) > 0 x (0, 1].
8
. N : f(x) G(x) > F(x)
x (0, 1].
6
. N :
G(1)
F(1)
G(x)
F(x)
x (0, 1].
4
. :
x dt g(t)
dt t dt g(t) f(t) lim
5 x
0
x
02 x
0
0 x
2
+ .
7
4
5
( )
1. ( , , ). .
2. , . .
.
3. . 4.
. , .
5. .
6. : (3) .
7. : 10.30 .
K
5
1
3 2007
:
: (4) 1o
A.1 f x0, .
10
.2 Rolle ;
5
B. , , , , , . . f()
f . 2
. f, g, g [,],
=dx)x(g)x(f
dx)x(f dx)x(g . 2
. f ,
=
x
dt)t(f f(x) x. 2
1
2
. f (,), (,) = = . )x(flim
x + )x(flimx 2
. f, g . f, g f(x) = g(x) x , f(x) = g(x) x.
2
2
++
3
3
f(x) = ex e lnx, x > 0.
. f(x) (1, +).
10
. f(x) e x > 0. 7
.
dt)t(fdt)t(f dt)t(f4
2
2x
3x
2x
1x
2
2
2
2 += ++++
(0, +). 8
4
z1 = +i
1
12
z2
z2z
+= ,
, IR 0. z2 z1 IR .
. z2 z1 = 1. 9
. z1 .
6
. >0,
z
2
1z
1
. 0 )i1z()i1z( 20120
1 =+++ 10
3
4
( ) 1. (,
, ). .
2. , . .
.
3. . 4.
. , .
5. . 6. : (3)
. 7. : 10.00 .
K
4
1
24 2008 :
: (5)
1o
A.1 f(x) = xln , x* * :
( )x
1xln =
10
.2 f [,];
5
B. ,
, , , , . . f:A 11,
f1 : )A(fy y,))y(f(f A xx,))x(f(f 11 ==
2
. f f .
2
1 5
2
. z2+z+=0 ,, 0 , .
2
. f ,
f( x ) > 0 x.
2
. A f ,,
+= f(x)dx f(x)dx f(x)dx 2
2
z w
3i)(3wi)(1w 6z)22i( ==+ :
. z .
6
. w .
7
. w
6
. wz 6
2 5
3
3
=>=
0x , 0
0x,lnx x f(x)
. f 0. 3
. f .
9
.
x
ex = . 6
.
f(x+1)>f(x+1)f(x) , x > 0 .
7 4
f
+= 203 45f(t)dt 3x)10x(f(x) .
f(x)=20x3+6x45 8
3 5
4
. g
.
h
h)(xg(x)glim(x)g
0h
= 4
. f () g ()
45f(x)h
h)g(x2g(x)h)g(xlim
20h+=++
g(0)=g(0)=1,
i. g(x)=x5+x3+x+1 10
ii. g 11 3
( )
1. ( , , ). .
2. , . .
.
3. .
4 5
5
4. . , .
5. .
6. : (3) .
7. : 10.30 .
K
5 5
1
3 2008 :
: (4)
1o
A. [, ]. G f [, ],
= )(G)(Gdt)t(f 10
. ;
5
. , , , , . . 11,
. 2
. f , f , .
2
.
dx)x(f
1 4
2
xx xx.
2
. , , :
+i=0 =0 =0 2
. (, x)(x, ) . :
`
0==
)(f(x)limf(x)lim
oo xxxx
``
2
2
2
3i1z1
+= z2+z+=0, .
. =1 =1.
9
. . 1z31 = 8
. w, :
1zzw 1= 8
2 4
3
3
.x,xln x f(x) 2 02 >=. : f(x)1 x>0.
6
. f.
6
.
0x
0x
,k
,)x(f
xln
)x(g
=
>
=
i. k g .
6
ii. 2
1k = , g ,
, (0, e).
7
4 f [0, +) f(x) > 0 x 0. :
F(x) = , x [0, +), x0 dt f(t) = x dt)t(ft
)x(F)x(h
0
, x (0, +).
3 4
4
. =+10 1 1)(Fdt)]t(F)t(f[e t 6
. h (0, +).
8
. h(1)=2, :
i. < 20 tf(t)dt 2 dt f(t) 20 6
ii. )(Fdt)t(F 12
11
0 = 5
1. ( , , ). .
2. , . .
.
3. . 4.
. , .
5. . 6. : (3)
. 7. : 10.00 .
K
4 4
1
( ) 20 2009
:
: (5)
1o
. f . f x 0)x(f = , f .
10
. f x0 ;
5
. , , , , . . z1, z2 ,
2121 zzzz = 2
. f () x0A, f(x)f(x0) xA
2
1 5
2
. 1x
1xlim
0x=
2
. f .
2
. f [, ] f(x)
3
. w
02
z12ww =+
. 0z
8
3
,1x ),1xln( (x)f x >+= 10 >
A. 1)x(f ,1x > =e 8
. =e,
. f .
5
. f ]0,1( ),0[ +
6
. , ),0()0,1( + , 0
2x
1)(f
1x
1)(f =+
(1, 2) 6
3 5
4
4
f [0, 2]
( ) 0dt)t(f2t20
=
= x0 ],2,0[x,dt)t(ft)x(H
=+
=
0 x,
t
tlim
],(x,dt)t(fx
)x(H
)x(G
t
x
2
2
0
0
116
203
. G [0, 2].
5
. G (0, 2)
2x0,x
)x(H)x(G
2
5
1. ( , , ). .
2. , . .
.
3. . 4.
. 5.
. 6. : (3)
. 7. : 10.00 . .
K
5 5
1
9 2009 :
: (5)
1o
A. f(x) = x . f (0 , + ) :
x2
1)x(f =
9
B. f xo . f xo ;
6
. ,
, , , . . z
( ) z)z( = 2
. f 1-1, f .
2
1 5
2
. f(x) = 0 f(x) < 0 xoxx
lim o
oxxlim )x(f
1 = +
2
. f(x) = x. H f 1= }{ 0xx =
= )x(f -x
12
2
. f, ,
)x(f dx = f(x) + c, x c .
2
2
z :
( ) ( ) 08zi2zi2 =++ . N
z = x+yi .
10
2 5
3
. N .
1z
2z
8
.
21 z,z
40zzzz 2121 =++ 22
lim
7
3
= f(x) ln[(+1)x2+x+1] - ln(x+2), x > -1 -1 . ,
f(x) . +x 5
. = -1 . f
. 10
. f
6
. f(x) + 2 = 0 0
4
3 5
4
4
f: [ ]2,0
x2exk)x(f4)x(f4)x(f =+ , 0 2x )0(f2)0(f = , f (2) = 2 f(2)+12 e4, f(1) = e2
k .
.
g(x) = 3x2- x2e
)x(f2)x(f , 0 2x
Rolle [0,2].
4 . (0,2) ,
)(f4)(f + = 6 e2 + 4 )(f 6
. k = 6 g(x) = 0 x [0,2].
6
. 0,ex)x(f x23= 2x 5
.
dxx
)x(f2
1 2 4
4 5
5
1. ( , , ). .
2. , . .
.
3. . .
4. . , .
5. .
6. : (3) .
7. : 10.00 . .
K
5 5
1
1 4
( ) 19 2010
:
: (4)
A1. f . F f , :
G(x)=F(x)+c, c
f
G f
G(x)=F(x)+c, c 6
A2. x=x0 f ;
4 A3. f
. f ;
5 4. ,
, , , .
) +i +i .
2
2 4
) f . f , .
) f (,), (,),
)x(flimB)x(flimAxx + ==
) (x)=x, x ) 0)x(flim
0xx
3
3 4
f(x)=2x+ln(x2+1), x 1. f.
5 2. :
( ) + +=+ 1x 1)2x3(ln2x3x2 4 22 7
3. f f .
6 4.
=1
1
dx)x(xfI
7
f: x :
f(x)x f(x)x =3+ x
0
dtt)t(f
t
1. f
f(x)=x)x(f
)x(f
, x 5
2. g(x)= ( )2)x(f 2xf(x), x, .
7
4
4 4
3.
f(x)=x+ 9x2 + , x 6
4.
+++ < 2x
1x
1x
x
dt)t(f dt)t(f , x
7
1. ( , , ). .
2. , . .
.
3. . 4.
.
5. .
6. . 7. : (3)
. 8. : 10.00 . .
K
1
7 2010 :
: (5)
A1. f(x) = x, x, )x( = x
8
A2. f
[,] ;
4
A3. f
x0A () , f(x0); 3
4. , , , , .
) f(x) = x, > 0, ( ) 1= xx x ) fog gof,
fog = gof
) += )x(flimxx0
, 010
= )x(flimxx
1 5
2
) f
[,] f(x) x[,],
0
dx)x(f 0
) zC zzz =2 10
z1, z2
z1 +z2 = 2 z1 z2 = 5 B1. z1, z2
5
B2. w
221
22
21 zzzwzw =+
w
(x+1)2 + y2 = 4
8
B3. w 2
2 Re(w) + Im(w) = 0 6
2 5
3
B4. w1, w2 w
2 421 = ww , 221 =+ ww
6
f(x) = (x2)lnx + x 3, x > 0
1.
f
5
2. f
(0,1] [1, + ) 5
3. f(x) = 0
.
6
4. x1, x2 3 x1 < x2,
(x1, x2) , f() f() = 0
f ( ))(f, .
9
3 5
4
f: f(0) = 1 f(0) = 0
1. f(x) 1 x 4
2. +=+
x
xdt)xt(fx
limx 3
1
0
3
0
6
f(x) + 2x = 2x ( )2x)x(f + , x , :
3.
f(x) = x2xe 2, x
8
4.
h(x) = +2xx
dt)t(f , x 0
5
1. ( , , ). .
2. , . .
.
3. . 4.
.
5. .
6. . 7. : (3)
. 8. : 09.30 . .
K
5 5
1
1 4
( ) 16 2011
:
: (4)
A1. f x0 . f x0 , : f (x0) = 0
10
A2. f . y=x+ f + ;
5 A3. ,
, , , .
) z 0 z0=1 ) f:A 1-1,
Ax,x 21 : x1x2, f(x1) f(x2)
) x1={x |x=0} : x
1)x(
2 =
) : 1x
xlim
x=+
2
2 4
) C C f f1 y=x xOy xOy.
10
z w iz 3 , :
2i3zi3z =++ i3z
1i3zw +=
B1. z
7
B2. i3z
1i3z =+
4
B3. w 22 w
8
B4. : zwz = 6
f : , , ( ) 0)0(f0f == , :
( ) )x(fx)x(f1)x(f)x(fex +=+ x.
3
3 4
),xeln()x( x =1. : f x 8
2. f .
3 3. f
.
7 4. = x
)xeln( x
2
,0
7
f, g : , x :
i) f(x)>0 g(x)>0
ii) += x0
t2
x2dt
)tx(g
e
e
)x(f1
iii) += x tx dt)tx(f ee )x(g0
2
2
1
1. f g
f(x) = g(x) x. 9
2. :
f(x) = ex, x 4
4
4 4
3. :
x
f
)x(flnlim
x 10
5
4.
= x dt)t(f)x(F1
2
xx yy x=1.
7
( ) 1. ( ,
) . .
2. . . .
3. . 4.
. , .
5. . 6. . 7. : (3)
. 8. : 10.00 . .
1
1 5
6 2011
:
: (5)
A1. f(x)=x x
= x )( x 10
A2. f, . f .
5
3. , , , , .
) z=+i, , z z=2
) f x0A () f(x0), f(x) f(x 0) xA
) f
, 1-1 .
2
2 5
) 0 f(x)>0 x= )x(f0xxlim 0,
+= )x(f1
lim0xx
) f x0 .
10
z, w,
:
iz =1+Im(z) (1) w( w +3i)=i(3 w +i) (2)
B1. z
y=4
1x2
7
B2. w (0,3) =2 2 .
7
B3. , z, w z =w.
5
B4. N , , u ,
3
3 5
,,, .
6
y= x , x . 0 (0,1) xy , .
t, t 0 m/min16)t(x = 1. ,
t, t : 0 x(t)=16t
5
2. (4,2) , , .
6
(0,1)
(4,2) y= x
y
O x
4
4 5
3. .
6
4. t0 (0,4
1),
d=() .
8
xy.
f: , 3 , :
i) )0(f1x
)x(flim
0x+=
ii) f(0) < f(1)-f(0)
iii) 0)x(f x 1.
f x0=0.
3
2. f . 5
g(x)=f(x)x, x , : 3. g
)x(xg
x
0xlim
6
5
5 5
4 . >2 20
dx)x(f
5
5. g, xx x=0 x=1 ()=e
2
5,
10
dx)x(f
(1,2) ,
0
dt)t(f =2
6 ( )
1. ( , ) . .
2. . . .
3. . 4.
. , .
5. . 6. . 7. : (3)
. 8. : 18.00
K
1
1 4
( ) 28 2012
:
: (4)
A1. f . x , f
0(x)f >
7
A2. f [, ];
4 A3. f .
f x0A ; 4
A4. , , , , .
)
) f 1-1, y f(x)=y x
) = + , f(x)
2
2 4
) ,x
1)(x
2= x{x |x=0}
) ,(x)g(x)dxf[f(x)g(x)](x)dxgf(x)
+= g,f
[,] 10
z w :
4=+z +1z 22
1_ (1)
12= w 5_w (2)
B1. z = 1
6
B2. z1, z2 z 2=zz 21
_ , .zz 21 + 7
B3. w
14
y
9
x
22 =+ w
6
B4. z,w (1) (2) :
1 wz 4 6
3
3 4
f(x)=(x1) nl x1, x>0 1. f
1=(0,1] 2=[1,+). f
6
2. x>0 .
,ex 20131-x = 6
3. x1, x2 x10, x=e xx
7
f : (0,+) , x>0 : f(x) 0
e
xx f(t)dt
21xx
1
2 + xx = nl f(x)edt
f(t)
tnt
x
1
+ l
1. f .
10
4
4 4
f(x) = ex( nl xx), x>0, : 2. : ( )( ) ( ) ( ) + xfxf
1xflim 2
0 x
5
3. nl xx1, x>0,
( ) dt, f(t)xF x = x>0,
>0, ( 2). :
F(x) + F(3x) > 2F(2x), x>0 ( 4).
6
4. >0. (,2) :
F() + F(3) = 2F()
4 ( )
1. ( , ) . .
2. . . .
3. . 4. .
, .
5. . 6. . 7. : (3)
. 8. : 10 .30 . .
1
1 4
14 2012
:
: (4)
A1. f (, ), x0, f . f(x)>0 (, x0) f(x)
2
2 4
) f [, ]. G f [, ],
= )(G)(Gdt)t(f 10
z, z1,
w=1z
1z
+
.
:
B1. 1z = 7
B2. 4
z
1z
.
6
B3.
+21 z
1
z
1 (z1+z2)4, z1, z2
z
6
B4. u,
uui=w
iw, w0, x2y2=1
6
f: , :
xf(x)+1=ex , x .
3
3 4
1. f(x)=
=
0x,1
0x,x
1ex
6
2. o f1 .
6
3. f ( ))0(f,0 . , f ,
2f(x)=x+2, x .
8
4. ( )[ ])x(fn)x(x llnlim0 x+
5
f:A =(0,+ ), :
f(A)= ( ]0, f (0,+ ), 2f(x)+ ,2dt
t
1tfee
x
1x
x
1
)t(f)x(f)t( +
+=
+ x>0
F(x)= dtfx
1)t( , x>0
1. f(x)=
+1xx2
n2
l , x>0
8
4
4 4
2. F ( ))x(F,x 00 , x0>0, . (x0, ) >x0, F M( ))(F,
: F() x(1)y+2012 (1)=0 6
3. >1,
[ ]0
3x
)1x()1(
1x
x)(f)1()(F 35 =++
+
, x, (1,3) 5
4 .
0x,dt)t(ftdtx
tf
x
1
x
x
2
>
6
( ) 1. ( ,
) . . 2.
. . .
3. . 4. .
, .
5. . 6. . 7. : (3)
. 8. : 18 .30
K
2003
.... 2003
-
1
. ) ( )f x x= . f
R : ( )'f x x= . 8
) ,f g :
( ) ( )' 'f x g x= x .
c : ( ) ( )f x g x c= +
x 5
) f . :
f Ax 0
. 3
. .
) RAf : 1f , f
.
) f 0x ( )0 0f x > , ( ) 0f x >
x 0x .
) f ],[ ,
( )0 ,x ( )0' 0f x > .
) f
. ( )'' 0f x > x .
) f [ ],a ( ) 0f x ( ) 0a
f x dx
> ,
[ ]0 ,x ( )0 0f x > .
) f ],[
( ) 0a
f x dx
= , f .
9
2003
2
( )( )ln
, 0ax
f x ax
= > .
. f (1, ( )1f )
0x y = , .
5
. 1a = :
) f .
5
) .
7
) : ( ) ( )1
1
+
> + 8
8
3
f [ ], 0 < <
z a i= + )()( fifw += ( ) 0f .
. :
) ( )
1
1
1
i zz
f i w
+ =
+
( )f a a=
5
) z iw= ,z w 0
, .
5
. 2 2 2
z iw z iw = + . :
) ( ) ( ) 0a f f =
4
) ,z w 0 .
3
) ( )0 ,x a ,
f ( )( )0 0,M x f x 0(0,0).
8
2003
4
f ( )''f x R :
( ) ( ) ( ) ( )0 1
2
0 0
1 '' 2 ' 4
x
x
t f t dt t f t dt x t f x dt+ = Rx ,
( )0 0f = ( )' 0 2f = .
) Rxx
xxf
+
= ,1
2)(
2
10
) ( )E a
f , xx 0x = 0x a= > .
a 10
/ sec3cm ,
( )E a , 3a cm= .
5
) g :
( ) ( )2g x x f x+ Rx .
(i) 2y x= +
g x+
5
(ii)
g , + 0x =
2x = , : ln 5E
5
2004
.... 2004
-
1
. :
f , [ , ] .
f [ , ]
f ( ) f ( )
, f ( ) f ( ) ,
x 0 ( , ) ,
f ( x 0 ) =
5
. f , ,
.
0 1 2 3
1
y
x x
y
y = f (x)
, I 1 , I 2 , I 3 . 3
10
I f (x) dx= 2
=3
02 dx (x)' f I 2
3
30
I f ''(x) dx= 3
2004
. .
1 . x
x lim
0x
2 .
x
1xlim
0x
3 . lnx lim0x +
4 . x
x e
1 lim
.
. 0
. 1
. +
8
. f ( x ) = x, IN -{0, 1}. , f
IR f ( x ) = x - 1
. 5
2
.
f , g IR
f ( x ) g ( x ) = 1 , f ( x ) 1 x IR...
2xf(x)
2g(x)L
x
lim
+=
+
,
0
0
.
. i ) L . 6
i i ) f g
+ . 6
. g IR . 6
. : f ( x ) g ( x ) = x + 4 x IR . 7
2004
3
x IR. +
=
x
0 t
dt e
2g(x) , > 0
z = g ( x ) + x i z + i | z 1 | .
. , i ) g i i ) z
g- 1
. 4
. , :
. R e ( z ) I m ( z ) , x IR 7
. = 1 . 7
.
12
2 t t 0 0
1 1 1 1 dt dt
1 e e e 1 e<
1 2,x x :
1 2x x= ( ) ( )1 2f x f x=
2. ( ) ( )0 0
lim limx x x x
f x g x
< ( ) ( )f x g x< 0x .
3. f [ ],a ( )0 ,x a
( )0 0f x = , ( ) ( ) 0f a f < .
4. f [ ],a ,
( )0 ,x a ( )0' 0f x < .
5. ( ) 0a
f x dx
= f
[ ],a , ( ) 0f x [ ],x a . 10
2
( ) ( )2 ln 2 , 0f x x x x= >
) : ( )ln
' , 0x
f x xx
= > .
4
) ( )0
lim 'x
f x+
.
2005
OEE
EMATA 2005
2005
2
3
) f . 8
)
( )ln x
g x
x
= , xx 1
x
e
=
2
x e= .
10
3
z = ex
+ (x 1) i, x .
) : ( ) ( )Re Imz z> x .. 8
) ( )0 0,1x 2
2w z z i= + + .
8 ) z .
9
4
f ( )1
02
f =
( ) ( ) ( )' 'xe f x f x x f x+ + = x .
) f xex
xf+
=1
)( , x
( ) ( )f x f x x+ = x .. 7
) ( )limx
f x+
.
6
) ( )2
2
I f x dx
= .
6
) : ( )2
0
04
f x dx
.
6
2006
2006
1
1
. ) f 0x
. f
0x ,
: ( )0' 0f x = . 11
) 0
x x=
f ;
4 .
.
) :f 1 2,x x
: ( ) ( )1 2f x f x= 1 2x x= .
) ( ) ( )( )0
limx x
f x g x
( )0
limx x
f x
( )0
limx x
g x
.
) ( )0
limx x
f x
= + ( ) 0f x x
0x .
) f
,
( )'' 0f x x .
) ( ) 0a
f x dx
= a < ( ) 0f x =
[ ],x a . 10
2006
2006
2
2
z 1
z iw
i z
+=
+
z i .
) : w i
zw i
=
+
5
) 1z = w ,
xx.
6 ) : w z .
7
) f [ ],a ( ) 1f a >
( )z f a i= ( )w f i= . ( ) 0f x = (,).
7
3
( ) 1xf x e ax= 1a > . )
f ( )( )0, 0f . 4
) f .
8
) ( )a
f , ( )( )0, 0f 1x a= > .
i) : ( )2
12
aa
a e a = .
7
ii) ( )lima
a
+
.
6
2006
2006
3
4
f ( ) 0f x >
( ) ( )1
0
, ,g x t f xt dt t x= .
:
) ( ) ( )2
0
1x
g x t f t dtx
= 0x .
6
) g 0
0x = .
6
) ( ) ( )0
x
x g x f t dt < 0x > .
7
) ( ) ( )2 1
1 0
3t f t dt t f t dt = ( )1,2
: ( ) ( )2g f = . 6
OE
E
EM
ATA 2
007
2007
1
1
&
1 . ) Fermat.
4
) f, ( )' 0f x > .
f .
9 .
.
1. Af : ( )f x
x , f .
2. ( )0
lim 0x x
f x
= , ( )f x 0x
( )0
lim 0x x
f x
= .
3. f ( ) ( ) 0f a f < ( ) 0f x
( ),x a , f [ ],a .
4. ,f g
( ) ( )' 'f x g x= x , ( ) ( )f x g x=
x .
5. f ,
: ( ) ( )f x d x f x d x = .
6. ,f g [ ],a
( ) ( )f x g x< [ ],x a , ( ) ( )a a
f x d x g x d x
OE
E
EM
ATA 2
007
2007
2
2
2
xeaxxf += )()( 2 , x . 2 2y x= +
f ( )( )0, 0f : ) : 2a = .
7
) f .
6
) : i) ( )lim
x
f x
ii) ( )limx
f x+
6
) ( ) 2007f x = .
6
3 ,z w 0z w :
z w z w+ = .
:
) ( )Re 0z w = . 6
) z
w
.
5
) ,z w
, 0.
7
) f [ ],a 0 a < <
( ) ( ),z a i f a w f i = + = ( ) ( )'x f x f x =
( ),a .
7
4
( )2
0
1
1
x
g x d tt
=
+ t,x .
) g .
4
) : ( )2
1
xg x x
x
+ 0x
7
) : ( ) ( ) 0g x g x+ = x .
6
OE
E
EM
ATA 2
007
2007
3
3
) g , xx 0, 1x x= =
( )1
1 ln 22
g = ..
8
2008
2008
1
1
'
&
1
. . f, g .
f, g f(x) = g(x) ,
c , x :
f (x) = g(x) + c
6
. f (x) = x, IN{0, 1}
IR :f(x) = x1
5
. 21
z,z .
() ():
. z1 z2
.
2
. : 2121
zzzz +=+
2
. : 212121
zzzzzz ++
2
. 21
zzzz = 21
zz
(z1) (z2).
2
2008
2008
2
2
4 10 x
4
y
A
B
. F(x) = dtf(t)x
0 , f
. () = 36 ..
:
. F(0) = . F(4) = . F(10) =
6
2
f
+
>+=
0x1,1)x(
0x,xf(x) , IR
. , f .
6
. , f x0 = 0.
8
. f 1-1.
3
. = 1 = 2, dxf(x)
2
.
8
3
f f (x) =x
e1e
, xIR .
. i. .
4
ii. f (x) =x
ex1x e1)(e + , f
.
5
. f.
6
. f.
4
.
f (x), xx, yy x = ln2
1.
6
2008
2008
3
3
4
f, g: IR IR x
:
=x
0
x
1
dtg(t)x2dtf(t) (1) g(x) 0 (2)
:
. f x0 = 0 f (0) = 2g(0)
6
. g(x) < 0 xIR
5
. 0
1
x
1
dtf(t)dtf(t) xIR
7
. H f (x) = 2g(x) + 2 (0, 1). 7
2009
2009
1
1
'
&
1
. f x0
. f x0
, : f(x0) = 0
9 . 1. = x +
f +;
3
2. f [, ];
3 . ()
():
1. 0
00
lim ( ) lim ( )x x h
f x l f x h l
= + =
2
2. 0
2009
2009
2
2
2
z + z
1 = 1, z C z1, z2 . :
A. z1z2 = 1 z1
3 =1.
4
B. (z1
2009 + z2
2009) R.
4
. z18 +
10
2z
1 + 1 = 0
4
. f(x) [0,1]
f(0)-2= 1 2
2 1
z z
z z
+ f(1)= 1 2
1 1 3
2 2 2z z+
x0 (0,1), f(x0)=3x0 2.
7
. w = 2z1 + 2z2 , z1 z2 , .
6
3
f(x) = x + 2 +2lnx .
. .
6
. f.
6
. 2
ln)(
+
=
x
xxxg x0 > 0 :
g(x) g(x0) x > 0.
7
. x > 2 : f(x 2) < 2f(x + 1) f(x + 4).
6
2009
2009
3
3
4
f (0, +)
:
f (x
1) =
x
e
x 1+ f(1) =
e
1
. f(x) = xe -1/x
.
8
. 1. f(x) x = 1.
2
2. 2
1
2( )f x dx
e> .
7
. g(x) = 3
)(
x
xf, (t)
Cg, xx x=1 x=t t> 1.
5
. E(t)limt +
.
3
2010
1
1
' .
&
1
. f [, ]. G
f [, ] , ( ) ( ) ( )f t dt G G
= . ( 10)
B.1. f x0
.
( 3)
B.2.
( )( )0 0,x f x f. ( 2)
. .
) 2 2
1 20z z+ =
1 2, Cz z
1 2z =z =0 .
) ( )g x x0 ( )0
limx x
g x
= ( )limy
f y l
=
( )( )0
limx x
f g x l
= .
) f [, ] ( )f , ( ) 0f = .
) f
, ( ) 0f x > .
) f [ ]2, 5 ( ) 0f x [ ]2, 5 ,
( )2
5
0f x dx . ( 10)
2010
2
2
2
z, w 1 2
1
w
z
w
+=
w
( )1,0 = 1.
) z (0, 0) = 1.
( 6)
) 1z = (1) 1 2 3, , z z z
(1) :
i) 2 3 1 31 2
3 1 2
z z z zz z
z z z
+ ++= + + .
( 7)
ii) z1+z2+z3=0 :
31 2
2 3 1
3Re
2
zz z
z z z
+ + =
( 7)
) (): 3 4 12 0x y+ = .
w ().
( 5)
3
( ): 0,f R+ , x > 0
( )( )
1
1f x
xx f x
e
+ =
+ ( )1 0f = .
) ( ) xg x e x= + 1-1.
( 2)
) ( ) lnf x x= x > 0. ( 6)
) ( )( ) 1f x
h xx
=
.
( 6)
2010
3
3
)
x x
x x
e e
=
0,2
x
.
( 5)
) h 1 2,x x
2 10x x> >
( ) ( )2 15
2 1
1
2
h x h x
x x e
.
( 6)
4
:f R R ,
3 1
( ) 2 6
x u
f t dt du x x R . :
)
3
1
( ) 2f t dt= . ( 7)
) f (0, f (0))
4 3 0x y+ =
2 3
0
40
( )
lim
x
x
t f t dt x
x
.
( 5)
) 1x ( ) 0f x >
1
( ) ( )
x
h x f t dt= , x > 1
( )( )
1
h xh x
x >
.
( 7)
) ( )1,3 , ( ) 3 2f + = .
( 6)
M
2011
1
1
'
&
1
. , , . : , , , , ) ,
7
. ;
4
. Rolle.
4
. () ().
1. z 2. 2
.
. 2
3. f : A R g : B R ,
, o .
2
4. , .
2
5. , , , . 2
M
2011
2
2
2
f : R R
4 12" " 1 , xR. R, = 1 . . i. " 1.
5
ii. . 5
.
lim
&
5
. i. 0, 1.
6
ii.
. 4
3
f : R R
x x 1, xR. . :
). *22 , 1 2 0 1 1
4 ii. 0, 1 , : x x 1
7
. , , - 2
, IR. i. .
/
0 . 6
ii. :
1 123 &
8
M
2011
3
3
4
. - 1, IR. 1;
3
. : 0, 0, . - 0 z, :
5 6
5 6
||2 5 66
8 1,
9 0. :
:. ).
; - 0, - 0. 5
ii. , - 0. 4
. . 5
. . 4
. (0, , , ,
2012
(....)
2012 _3.3()
: 1 3
:
: &
:
: . 11 2012
A1. , f x0, .
5
A2. f
x0;
4
A3. N f (x) = x, > 0 R.
f (x) = x ln
6
A4. :
i. 11, .
2
ii. i4 + 3
= i, .
2
iii. 0)x(flim0xx
>
, 0)x(f > x0.
2
iv. x, y y = f (x),
f x0, o y x x0 y = f (x0).
2
v. f ,
x0 , f (x0) 0, f.
2
2012
(....)
2012 _3.3()
: 2 3
f(x) = ex2
g(x) = lnx+2.
B1. fog gof .
6
B2. f f 1
.
6
B3. ex2
= lnx + 2 , ,
(e2
, 2).
6 B4. :
)x)(fog(
)x(glim
)x)(gof(
)x(flim
xx +
= = 0
7
f: IR IR x IR 1
x
2t f (t) dt2(1+3 ) f (x) = e ,
IR {0}.
1. :
i. f (x)f x 2(x) ' f 2= , x IR.
4
ii. 2 2
1f (x) =
x + 3 , xIR.
4
2.
0
t f (t) dt .
4 3. f.
8
4. ,
f x = , :
|| 3
1 E
|| 4
1
2012
(....)
2012 _3.3()
: 3 3
f IR f (0) = 2,
x+2
x 2
f (x) 2elim = 1
x + 2
f (x) < 0, xIR.
:
1. f (2) =1 f (x) x + 4, xIR.
6
2. f x0(2, 0).
6
3.
( )2(x 5)
0f f (t x)dt = f (0)
IR x = 5.
7
4. z
f(|z + i|) f(|z| + 1)
.
6
1
1 4
( ) 27 2013 - :
: (4)
A1. f [ ], . G
f [ ], , :
( ) ( ) ( )
f t dt G G =
7 A2.
( ...) 4
A3. f [ ], ;
4 A4. ,
, , , .
) 0z z , >0 = ( )0K z 2 , 0z, z .
) ( )0x x
lim f x 0
< , ( )f x 0< 0x
) : x x x
) : x 0
x 1lim 1x
=
) f f .
10
2
2 4
z :
( ) ( )z 2 z 2 z 2 2 + =
B1. z , ( )K 2,0 1= ( 5)
, z , z 3 ( 3)
8
B2. 1 2z , z 2w w 0+ + = , w , , ,
( ) ( )1 2z z 2 =Im Im :
4= 5= 9
B3. 0 1 2 , , 1. v :
3 22 1 0v v v 0+ + + =
:
v 4<
8
f,g : , f :
( )( ) ( )( )f x x f x 1 x+ + = , x ( )f 0 1=
( )2
3 3xg x x 12
= +
3
3 4
1. :
( ) 2f x x 1 x= + , x 9
2.
( )( )f g x 1= 8
3. 0x 0, 4
, :
( ) 0 00
0
x 4
f t dt f x x4
=
8
( )f : 0, + :
f ( )0, + ( )f 1 1=
( ) ( )
h 0
f 1 5h f 1 hlim 0
h+
=
( ) ( )x
f t 1g x dt
t 1
= , ( )x 1, + 1>
:
1. ( )f 1 0 = ( 4), f 0x 1= ( 2).
6 2. g ( 3), ,
2 4
2 4
8x 6 2x 6
8x 5 2x 5g(u)du g(u)du
+ +
+ +
> ( 6) 9
4
4 4
3. g ,
( ) ( ) ( )( ) ( )x
f t 1 1 dt f 1 x , x 1
t 1
= >
. 10
( )
1. . - . - . .
2. . . .
3. . , , .
4. . 5. : (3)
. 6. : 10.00 . .
K
1
1 5
13 2013 :
: (5)
A1. f 0x ,
f . 7
A2. Fermat. 4
A3. f . f ;
4 A4. ,
, , , . ) z z z=
( 2)
) f 11 , f .
( 2)
) ( )0x x
lim f x
= , ( )( )0x x
lim f x
= +
( 2) ) f , g 0x
:
( ) ( ) ( ) ( ) ( ) ( )0 0 0 0 0f g x f x g x f x g x = ( 2)
) f
, f . ( 2)
10
2
2 5
z, w
22x w 4 3i x 2 z , x =
, x 1=
B1. z
1 1, =
w
(4,3) 2 4 =
8
B2. N , .
5
B3. z , w 1 :
z w 10 z w 10+
6
B4. z 1 , :
22z 3z 2zz 5 =
6
f : :
( ) ( )22x f x x f (x) 3 f (x) + = x
( ) 1f 12
=
3
3 5
1. :
( )3
2xf x ,
x 1=
+ x
f 6
2. f 1.
4
3. :
( ) ( )2 3 2 2f 5(x 1) 8 f 8(x 1)+ + 7
4. , , ( )0, 1 , :
( ) ( ) ( )3
2 3
0
f t dt 3 1 f
= 8
[ )f : 0, + , [ )0, + , :
( ) ( )( )( )
x 2
1
u
1
f t 1f x x dt du
f t
= + x 0>
( ) ( )f x f x 0 x 0> ( )f 0 0=
:
( )( )
f xg(x)
f x
= x 0> ( ) ( )( )3h x f x= x 0
4
4 5
1. N :
( ) ( ) ( )( )2f x f x 1 f x + = x 0> 4
2. . f f ( )0, + ( 4)
. ( )f 0 1 = ( 3) 7
3. g ( )0, ,+ :
. ( )g x 2 x x ( )0, + ( 2)
. ( ) ( )1
0
2 x f x dx 1
5
5 5
3. . , , .
4. . 5. : (3)
. 6. : 18:00
K
2013
(....)
2013 _3.3()
: 1 3
:
: &
:
: . 30 2013
: 3
1. f (, ),
x0, f .
, ( )f x >0 0 0
( , ) ( ,)a x x , 0
( )f x
f (,).
9
2. . (x0, f(x0)) f.
3
. f, g , , f g ;
3
3. , , , .
) ,f g [,] ,
[ ]( ) ( ) ( ) ( ) ( ) ( ) = aa a
f x g x dx f x g x dx f x g x
) z *, v
v
z z = .
) 11 , .
) 0 1a< < lim loga
x
x
+
= + .
) * ( ) vf x x= * 1( ) vf x vx = .
10
2013
(....)
2013 _3.3()
: 2 3
z w :
2
( 2) 1 3z z i z+ = 2 ,w z i=
1. z.
z ;
7
2. zz z
.
6
3. z 2= zz
0)Im( >z , 2013
2
zz.
6
B4. w z w
z (0,1).
6
, 0
( ) 1
ln , =0
x
xx
f x e
a x
=
.
1. (0, )a + f
1(0)
2f = .
7
= e.
2. . f . 6
. , .
6
3. 0
1 12
( ) 1 2013
x
x dtf t
=
+
(0, 1).
6
2013
(....)
2013 _3.3()
: 3 3
f, G F, [0, +) f G .
: f (0) = 1, G(0) = 0 x 0
( )f x > 0, '( ) 1G x > 0
( ) ( ) dx
F x f t t= .
1. ( ) 0F x ( )G x x x 0 .
5
2. [ ]0
lim ( ) lnx
F x x
(0, 1)
, ( )
( ) ln 0F
f
+ = .
7
3. , ,
[ ] [ ] 22
( ) ( ) ( ) ( ) ( ) ( ) 1f x F x f x G x G x x G x + = + , x 0.
:
. ( ) ( )F x G x x= , x 0.
7
. x0 >0, F GC , C
( )( ) ( )( )0 0 0 0, ,x F x x G x , yy ( 3)
, F G
C , C 0
x x=
( 3).
6