6. 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 : . . . ,
7. 2 . f x0 ,0f(x) xx lim 0 = .0f(x) xx lim 0 = 1 . f R ,
.dx)x(xf)x(xfdx)x(f = 1 . f(x) > 0 x,0f(x) xx 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 i 2 . 8 . z= 2 Arg(z) = 3 , 0, z f(13). 10 2 : .
. . ,
8. 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)xf(x) 2 x 0. 12 ) f, x = 0, x = 1 xx, )1(f 2 1 E 4 1 |z 1| 9
) f 8 4 f R, Rx),x(f )x(f=))x(f (+)x(f)x(f 2 1=)0(f 2=)0(f ) f 12
B) g [0, 1] x 0 2 1=dt )t(f+1 g(t) -x2 [0, 1] 13 : . . . ,
11. 1 29 2003 : : 1o A. , f x0, . 8 . ; 7 . , . . z _ z , zzz
== . 2 . f . f(x)>0 x , f . 2 1 : . . . ,
12. 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 : . . . ,
13. 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()0, f(3)= > , x0 (2,3) f(x0)=0. 6 :
. . . ,
23. 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 : . . . ,
24. 2 2 . C C f f1 y = x xOy xOy. 2 . f x0, f(x)lim)x(flim k
0xx k 0xx = , f(x) 0 x0, k k 2. 2 . f (, ) [, ]. 6 2 f: IR IR f(x)
= 2x + mx 4x 5x , mIR, 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). : . . . ,
25. 3 3 z 1 z + = f() z 1 z 2 2 + = f2 (), : . z= 1 11 . f2 ()
< f2 () 5 . x3 f() + f() = 0 (1, 1). 9 4 f [0, +) IR , += 2 1 0
2 dt2xf(2xt) 2 x f(x) . . f (0, +). 7 . f(x) = ex (x + 1). 7 . f(x)
[0, +). 5 . f(x)lim x + f(x)lim x . 6 : . . . ,
31. 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 : . . . ,
32. 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 : . . . ,
33. 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 . Cf, Cf (): 2005x 668 1 y += . 9 4 f:
IRIR, 2005 x x)x(f lim 20x = . . : i. f(0)=0 4 ii. f(0)=1. 4 : . .
. ,
34. 4 4 . IR , : ( ) ( ) .3 )x(fx2 )x(fx lim 22 22 0x = + + 7 .
f IR f(x)>f(x) x IR, : i. xf(x)>0 x0. 6 ii. . < 1 0
)1(fdx)x(f 4 ( ) 1. (, , ). . 2. , . . , . 3. . 4. . 5. : (3) . 6.
: 10.30 . K : . . . ,
36. 2 . H f() f . 2 . , x IR.1-xx 3x)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 : . . .
,
37. 3 3 1z,z,z 321321 zzz === .zz 0z 321 =++ . : i. 321321
zzzzzz == . 9 ii. 4zz 2 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 : . . . ,
64. 4 4 f [0, 2] ( ) 0dt)t(f2t 2 0 = = x 0 ],2,0[x,dt)t(ft)x(H
= + = 0x, t t lim ],(x,dt)t(f x )x(H )x(G t x 2 2 0 0 11 6 203 . G
[0, 2]. 5 . G (0, 2) 2x0, x )x(H )x(G 2