Transcript
• . ' ,

2 8 8 8

- 2015 - 2

Rolle

10( ) -

)37((((((((

.

• ( 1 - 4)

' 2.

.

, .

4 , 2010.

( 5 - 10)

:

37

.

. , .

6. Rolle.

- www.mathsteki.gr

- 1 -

• 1. f :! ! , , x'x 1, 2, 3. , , (1,3) , f () = 0 .

2. f :! ! , , f (0) =1 , f (1) = e f (2) = e

2 . , , g(x) = f (x)e

x + x 23x .

) , , C

g

(0,2) , Cg .

) , , (0,2) , f ()+ 2 = e .

3. f :! ! , . f x'x x1 x2 , x1 < x2 . , , ! , f

(3)() = 0 .

4. f : [1,3] ! , , f (1) = 2 , f (2) = n(2e

3) f (3) = n(3e4) . :

) g(x) = f (x) nx x - [1,3] .

) , , (1,3) , 2 f () = 1 .

5. f [0,1] , f (0) = 0 . : ) g(x) = f (x)x f (0)[f (1) f (0)]x

2 - Rolle [0,1] .

) , , (0,1) , f (1) f (0) =

12 f () .

• 6. f, [,] , f () = f () = f () = f () = 0 .

: ) g(x) = e

x [ f (x) f (x)] Rolle [,] .

) , , (,) , f () = f () .

7. f, [,] - (,) ,

f ()f ()

=f ()f ()

f (x) 0 , x ! .

x0 (,) , f (x0) f (x0) = [ f (x0)]2 .

8. , f, ! , f (1) = 1 , f (2) = 4 n2 , f (e) = e

21 .

(1,e) ,

f () =12

+ 2 .

9. f, ! , f (0) = 0 , f () = f () =

2 .

( ,) , f () = 2 .

10. f, ! , f (1) = f (1) = 1 , . (1,1) , f , ( , f ()) :y = 2x 3 .

• 1. f :! ! , , x'x 1, 2, 3. , , (1,3) , f () = 0 .

C

f x'x 1, 2, 3, f (1) = f (2) = f (3) = 0 .

f ! , f , f , :

f , , ! .

f , , ! .

. f [1,2] , [2,3] .

. f (1,2) , (2,3) .

. f (1) = f (2) = f (3) = 0 . Rolle ,

x1 (1,2) , f (x1) = 0 , x2 (2,3) , f (x2) = 0 .

. f [x1 ,x2 ] .

. f (x1 ,x2) .

. f (x1) = f (x2) = 0 .

Rolle, (x1 ,x2) , (1,3) ,

f () = 0 .

. , .

6. Rolle.

- www.mathsteki.gr

- 4 -

• 2. f :! ! , , f (0) =1 , f (1) = e f (2) = e

2 . , , g(x) = f (x)e

x + x 23x .

) , , C

g

(0,2) , Cg .

) , , (0,2) , f ()+ 2 = e .

) , , f, ' - , f () = 0 .

, x1 ,x2 (0,2) , g (x1) = g (x2) = 0 .

f ! , f , f , :

f , , ! .

f , , ! .

. g [0,1] , [1,2] .

. g (0,1) , (1,2)

. g(0) = f (0)e0 + 02 3 0 =11 g(0) =2 .

IV. g(1) = f (1)e112 3 1 = ee +13 g(1) =2 .

V. g(2) = f (2)e2 + 22 3 2 = e2 e2 + 46 g(2) =2 .

Rolle ,

x1 (1,2) , g (x1) = 0 , x2 (1,2) , g (x2) = 0 .

) g (x) = f (x)ex + 2x 3 .. g [x1 ,x2 ] .

. g (x1 ,x2) .

. g (x1) = g (x2) = 0 ().

Rolle, (x1 ,x2) , (0,2) ,

g () = 0 (1)

g (x) = f (x)ex + 2 , (1)

f ()e + 2 = 0 f ()+ 2 = e .

. , .

6. Rolle.

- www.mathsteki.gr

- 5 -

• 3. f :! ! , . f x'x x1 x2 , x1 < x2 . , , ! , f

(3)() = 0 .

C

f x'x x1 ,x2 , f (x1) = f (x1) = f (x2) = f (x2) = 0 .

f ! , f , f , f(3) , :

f , , ! .

f , , ! .

f , , ! .

. f [x1 ,x2 ] .

. f (x1 ,x2) .

. f (x1) = f (x2) = 0 .

Rolle, x0 (x1 ,x2) , f (x0) = 0 .

. f [x1 ,x0 ] , [x0 ,x2 ] .

. f (x1 ,x0) , (x0 ,x2) .

. f (x1) = f (x0) = f (x2) = 0 .

Rolle ,

1 (x1 ,x0) , f (1) = 0 , 2 (x0 ,x2) , f (2) = 0 .

. f [1 ,2 ] .

. f (1 ,2) .

. f (1) = f (2) = 0 .

Rolle, (1 ,2) , f(3)() = 0 .

. , .

6. Rolle.

- www.mathsteki.gr

- 6 -

• 4. f : [1,3] ! , , f (1) = 2 , f (2) = n(2e

3) f (3) = n(3e4) . :

) g(x) = f (x) nx x - [1,3] .

) , , (1,3) , 2 f () = 1 .

) , , f, ' - , f () = 0 .

, x1 ,x2 (1,3) , g (x1) = g (x2) = 0 .

f [1,3] , f , f , :

f , , [1,3] .

f , , [1,3] .

. g [1,2] , [2,3] .

. g (1,2) , (2,3)

. g(1) = f (1) n11 = 21 g(1) = 1 .

IV. g(2) = f (2) n22 = n2e3 n22 = n2 + ne3 n22 = 32 g(2) = 1 .

V. g(3) = f (3) n33 = n3e4 n33 = n3 + ne 4 3 = 43 g(3) = 1 .

Rolle ,

x1 (1,2) , g (x1) = 0 , x2 (2,3) , g (x2) = 0 .

) g (x) = f (x)

1x1 .

. g [x1 ,x2 ] .

. g (x1 ,x2) .

. g (x1) = g (x2) = 0 , ' ().

Rolle, (x1 ,x2) , (1,3) ,

g () = 0 (1)

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

, (1)

f ()+

12

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

. , .

6. Rolle.

- www.mathsteki.gr

- 7 -

• 5. f [0,1] , f (0) = 0 . : ) g(x) = f (x)x f (0)[f (1) f (0)]x

2 - Rolle [0,1] .

) , , (0,1) , f (1) f (0) =

12 f () .

) f [0,1] , f , f , : f , , [0,1] .

f , , [0,1] .

. g [0,1] - .

. g (0,1) - .

. g(0) = f (0)0 f (0)[f (1) f (0)]02 = f (0) g(0) = 0 .

IV. g(1) = f (1)1 f (0)[f (1) f (0)]12 = f (1) f (0) f (1)+ f (0) g(1) = 0 .

g Rolle [0,1] .

) g Rolle [0,1] , x0 (0,1) , g (x0) = 0 .

g (x) = f (x) f (0)2 [f (1) f (0)]x .

g (0) = f (0) f (0)2 [f (1) f (0)]0 g (0) = 0 = g (x0) .

. g [0,x0 ] .

. g (0,x0) .

. g (0) = g (x0) = 0 .

' Rolle, (0,x0) , (0,1) ,

g () = 0 .

g (x) = f (x)2 [f (1) f (0)] ,

f ()2 [f (1) f (0)] = 0 2 [f (1) f (0)] = f () f (1) f (0) =

12 f () .

. , .

6. Rolle.

- www.mathsteki.gr

- 8 -

• 6. f, [,] , f () = f () = f () = f () = 0 .

: ) g(x) = e

x [ f (x) f (x)] Rolle [,] .

) , , (,) , f () = f () .

) f [,] , f , f , : f , , [,] .

f , , [,] .

. g [,] - .

. g (,) - .

. g() = e [ f () f ()] = e 0 g() = 0 .

IV. g() = e [ f () f ()] = e 0 g() = 0 .

g Rolle [,] .

) g Rolle, (,) ,

g () = 0 (1)

g (x) = (ex ) [ f (x) f (x)]+ex [ f (x) f (x) ]

g (x) = ex [ f (x) f (x)]+ex [ f (x) f (x)] = ex [ f (x) f (x)+ f (x) f (x)]

g (x) = ex [ f (x) f (x)] .

(1) e [ f () f ()] = 0

e>0

f () f () = 0 f () = f () .

. , .

6. Rolle.

- www.mathsteki.gr

- 9 -

• 7. f, [,] - (,) ,

f ()f ()

=f ()f ()

f (x) 0 , x ! .

x0 (,) , f (x0) f (x0) = [ f (x0)]2 .

x0 = x

f (x) f (x) = [ f (x)]2 f (x) f (x)[ f (x)]2 = 0 [ f (x) ] f (x) f (x) f (x) = 0

f (x )0

[ f (x) ] f (x) f (x) f (x)f 2(x)

= 0 f (x)

f (x)

= 0 .

g(x) =

f (x)f (x)

, x [,] , x0 (,) , g (x0) = 0 .

f [,] (,) , f , f , :

f (,) .

f , , (,) .

. g [,] .. g (,) .

III. g() =

f ()f ()

, g() =f ()

f ().

f ()f ()

=f ()f ()

, ,

f ()f ()

=f ()

f () g() = g() .

Rolle, x0 (,) , g (x0) = 0 .

. , .

6. Rolle.

- www.mathsteki.gr

- 10 -

• 8. , f, ! , f (1) = 1 , f (2) = 4 n2 , f (e) = e

21 .

(1,e) ,

f () =12

+ 2 .

= x

f (x) =

1x 2

+ 2 f (x)1x 2

2 = 0 f (x)+1x

2x

= 0 [f (x)+ nx x 2 ] = 0 .

g(x) = f (x)+ nx x2 (1,e) ,

g () = 0 .

f ! , f , f , :

f , , ! .

f , , ! .

. g [1,2] , [2,e] .

. g (1,2) , (2,e) - .. g(1) = f (1)+ n11

2 g(1) = 0 .

IV. g(2) = f (2)+ n222 = 4 n2 + n2 4 g(2) = 0 .

V. g(e) = f (e)+ nee2 = e21+1e2 g(e) = 0 .

Rolle ,

x1 (1,2) , g (x1) = 0 , x2 (2,e) , g (x2) = 0 .

g (x) = f (x)+

1x2x .

. g [x1 ,x2 ] .

. g (x1 ,x2) .

. g (x1) = g (x2) = 0 .

Rolle, (x1 ,x2) , (1,e) ,

g () = 0 .

. , .

6. Rolle.

- www.mathsteki.gr

- 11 -

• 9. f, ! , f (0) = 0 , f () = f () =

2 .

( ,) , f () = 2 .

= x

f (x) = 2 x f (x)2 + x = 0 [ f (x)2x x ] = 0 [f (x)x2 x ] = 0 .

g(x) = f (x)x2 x ( ,) , -

g () = 0 .

f ! , f , f , :

f , , ! .

f , , ! .

. g [ ,0] , [0,] .

. g ( ,0) , (0,) - .. g() = f ()()

2 () = 22 + g() = 0 .

IV. g(0) = f (0)02 0 g(0) = 0 .

V. g() = f ()2 = 22 g() = 0 .

Rolle ,

1 ( ,0) , g (1) = 0 , 2 (0,) , g (2) = 0 .

g (x) = f (x)2x x .

. g [1 ,2 ] .

. g (1 ,2) .

. g (1) = g (2) = 0 .

Rolle, (1 ,2) , ( ,) , - g () = 0 .

. , .

6. Rolle.

- www.mathsteki.gr

- 12 -

• 10. f, ! , f (1) = f (1) = 1 , . (1,1) , f , ( , f ()) :y = 2x 3 .

f f () (),

f () =

f () = 2 f ()2 = 0 (1)

(1,1) , (1).

= x (1)

f (x)2 = 0 [ f (x)2x ] = 0 [f (x)x2 ] = 0 .

g(x) = f (x)x2 (1,1) ,

g () = 0 .

f ! , f , f , :

f , , ! .

f , , ! .

. g [1,0] , [0,1] .

. g (1,0) , (0,1) - .. g(1) = f (1)(1)

2 = 11 g(1) = 0 .

IV. g(0) = f (0)02 g(0) = 0 .

V. g(1) = f (1)12 = 11 g(1) = 0 .

Rolle ,

1 (1,0) , g (1) = 0 , 2 (0,1) , g (2) = 0 .

g (x) = f (x)2x .

. g [1 ,2 ] .

. g (1 ,2) .

. g (1) = g (2) = 0 .

Rolle, (1 ,2) , (1,1) ,

g () = 0 .

. , .

6. Rolle.

- www.mathsteki.gr

- 13 -