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1. f 0, f 3(x)2x f 2(x)+ x 2 f (x) = x 2 2x , x ! .

) f (0) , f (0) .

) imx0

f (x)x +1x

.

) f ! , , (x 2) [f (x)+ f (x)] = x , , (0,2) .

2. , x,

f 2(x)+ g 2(x) = x 6 + 2x 3 +1 (1), f, g

x0 =1 , :

) f (1) = g(1) = 0 . ) [ f (1)]2 + [ g (1)]2 = 9 .

3. f :! ! 1, f

3(x)+ (x 1)2 f (x)2(x 1)3 = 0 , x ! . f (1) = 1 .

4. f :! ! 1, x 2 f 3(x)+ f (x)+1 = x 3 , x ! . f (1) = 3 .

5. f, g 1 f

2(x)+ g 2(x) = (x 21)2 , x ! , [ f (1)]2 + [ g (1)]2 = 4 .

1. f 0, f 3(x)2x f 2(x)+ x 2 f (x) = x 2 2x , x ! .

) f (0) , f (0) .

) imx0

f (x)x +1x

.

) f ! , , (x 2) [f (x)+ f (x)] = x , , (0,2) .

) x ! , x = 0

f 3(0)2 0 f 2(0)+ 02 f (0) = 02 0 f 3(0) = 0 f (0) = 0 .

f (0) = im

x0

f (x) f (0)x 0

f (0) = imx0

f (x)x

(1)

x 0 , x 3 0 ,

f 3(x)x 3

2x f 2(x)

x 3+

x 2 f (x)x 3

=x 2 2x

x 3

f (x)x

3

2f (x)x

2

+f (x)x

=2x

x

imx0

f (x)x

3

2f (x)x

2

+f (x)x

= imx0

2xx

(2)

= im

x0

2xx

= imx0

2 2x2x

= 2 imx0

2x2x

, 2x = y .

x 0 , y 0 , = 2 im

y0

yy

= 2 1 = 2 (2),

(1), [ f (0)]3 2[ f (0)]2 + f (0) = 2 .

f (0) = ! 322 +2 = 0 (2)(2 +1) = 0 .

2 +1 > 0 , ! , 2 = 0 = 2f (0) = 2 .

. , , , , , , , .

. , .

1. .

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)

1

= imx0

f (x)x +1x

= imx0

f (x)(x 1)xxx

= imx0

f (x)x

x 1x

xx

1=

201

1= 2 .

) (x 2) [f (x)+ f (x)]+ x = 0 . g(x) = (x 2) [f (x)+ f (x)]+ x , x [0,2] , - g(x) = 0 , , (0,2) .

. f ! , ! . f - ! , g ! , [0,2] , .

. g(0) = (02) [f (0)+ f (0)]+ 0 = 2(0 + 2) = 4 < 0 .

III. g(2) = (22) [f (2)+ f (2)]+ 2 = 2 > 0 .

Bolzano, g(x) = 0 , , (0,2) .

2. , x,

f 2(x)+ g 2(x) = x 6 + 2x 3 +1 (1), f, g

x0 =1 , :

) f (1) = g(1) = 0 . ) [ f (1)]2 + [ g (1)]2 = 9 .

) (1) x ! , x =1

f 2(1)+ g 2(1) = (1)2 + 2 (1)3 +1 f 2(1)+ g 2(1) = 0 f (1) = g(1) = 0 .

) f 1 ,

f (1) = imx1

f (x) f (1)x (1)

= imx1

f (x)x +1

(2)

g 1 ,

g (1) = imx1

g(x)g(1)x (1)

= imx1

g(x)x +1

(3)

x 1 , x +1 0 (1)

f 2(x)+ g 2(x)(x +1)2

=x 6 + 2x 3 +1

(x +1)2

f 2(x)(x +1)2

+g 2(x)

(x +1)2=

(x 3 +1)2

(x +1)2

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1. .

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f (x)x +1

2

+g(x)x +1

2

=[(x +1)(x 2 x +1)]2

(x +1)2

f (x)x +1

2

+g(x)x +1

2

=(x +1)2(x 2 x +1)2

(x +1)2

f (x)x +1

2

+g(x)x +1

2

= (x 2 x +1)2 imx1

f (x)x +1

2

+g(x)x +1

2

= imx1

(x 2 x +1)2 (3)

(2)

[ f (1)]2 + [ g (1)]2 = [(1)2 (1)+1]2 [ f (1)]2 + [ g (1)]2 = 9 .

3. f :! ! 1, f

3(x)+ (x 1)2 f (x)2(x 1)3 = 0 , x ! . f (1) = 1 .

f (1) = im

x1

f (x) f (1)x 1

.

x ! , x = 1

f3(1)+ (11)2 f (1)2(11)3 = 0 f 3(1) = 0 f (1) = 0 .

f (1) = im

x1

f (x)x 1

(1)

x 1 , x 1 0 ,

f 3(x)(x 1)3

+(x 1)2 f (x)

(x 1)3

2(x 1)3

(x 1)3=

0(x 1)3

f (x)x 1

3

+f (x)x 1

2 = 0

imx1

f (x)x 1

3

+f (x)x 1

2

= imx1

0 (1)

[ f (1)]3 + f (1)2 = 0 .

f (1) = ! , 3 +2 = 0 (1)(2 ++ 2) = 0 1 = 0

= 1 f (1) = 1 , 2 ++ 2 12 4 1 2 = 7

( , ! ).

. , .

1. .

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4. f :! ! 1, x 2 f 3(x)+ f (x)+1 = x 3 , x ! . f (1) = 3 .

f (1) = im

x1

f (x) f (1)x 1

.

x ! , x = 1

12 f 3(1)+ f (1)+1 = 13 f 3(1)+ f (1) = 0 f (1) [f 2(1)+1] = 0 f (1) = 0 ,

f2(1)+1 > 0 , f(1).

f (1) = im

x1

f (x)x 1

(1)

x 1 , x 1 0 ,

x 2 f 3(x)+ f (x) = x 3 1

x 2 f 3(x)x 1

+f (x)x 1

=x 3 1x 1

x 2 f 2(x)

f (x)x 1

+f (x)x 1

=(x 1)(x 2 + x +1)

x 1

f (x)x 1

[x 2 f 2(x)+1] = x 2 + x +1

f (x)x 1

=x 2 + x +1

x 2 f 2(x)+1 (2) , x

2 f 2(x)+1 > 0 , f(x).

f 1, imx1

f (x) = f (1) = 0 , (2)

imx1

f (x)x 1

= imx1

x 2 + x +1x 2 f 2(x)+1

(1)

f (1) =12 +1+112 0 +1

f (1) = 3 .

. , .

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5. f, g 1 f

2(x)+ g 2(x) = (x 21)2 , x ! , [ f (1)]2 + [ g (1)]2 = 4 .

f 1, f (1) = im

x1

f (x) f (1)x 1

.

g 1, g (1) = im

x1

g(x)g(1)x 1

.

x ! , x = 1

f2(1)+ g 2(1) = (121)2 f 2(1)+ g 2(1) = 0 f (1) = g(1) = 0 .

f (1) = im

x1

f (x)x 1

, g (1) = imx1

g(x)x 1

.

x 1 , x 1 0 ,

f 2(x)(x 1)2

+g 2(x)

(x 1)2=

(x 2 1)2

(x 1)2

f (x)x 1

2

+g(x)x 1

2

=[(x 1)(x +1)]2

(x 1)2

f (x)x 1

2

+g(x)x 1

2

=(x 1)2(x +1)2

(x 1)2

f (x)x 1

2

+g(x)x 1

2

= (x +1)2

imx1

f (x)x 1

2

+g(x)x 1

2

= imx1

(x +1)2 [ f (1)]2 + [ g (1)]2 = (1+1)2 [ f (1)]2 + [ g (1)]2 = 4 .

. , .

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