Download pdf - Κ43-Θεώρημα Μέσης Τιμής (5)

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1. f: [!,"]! ! , f(!)=! ,f(")=" ,

0< !< " . :

) x0! (!,") , f(x

0)= !+ "!x

0.

) !1! (",x

0) , !

2! (x

0,#) , !f(!

1)" !f(!

2)= 1 .

2. f: [!,"]! ! , [!,"] (!,") ,

f(!)= 2" f(!)= 2" .

) f(x)= 2x, , (!,") .

) !1,!

2! (",#) , !f(!

1) " !f(!

2)= 4 .

3. f, [1,3] .

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

) [1,2] [2,3] .

) x0! (1,3) , !!f (x

0)= 0 .

4. f:[0,1]! ! , f(0)= 0 ,f(1)= 1 . -

:

) x0! (0,1) , f(x

0)=

1

2.

) !1,!

2! (0,1) ,

1

!f(!1

)+

1

!f(!2

)= 2 .

5. f: [!,"]! ! , [!,"] ,

x1,x

2,x

3! [!,"] . x

1,x

2,x

3f(x

1),f(x

2),f(x

3) -

, !! (",#) , !!f (!)= 0 .

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6. f: [!,"]! ! , f(!)> 0 ,f(")> 0 ,

f(!)< 0 , !! (",#) . :

) f(x)= 0 , , (!,") .

) x0! (!,") , C

fx

0, -

x'x.

) !! (",#) , !!f (!)> 0 .

7. f ! - C

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

8. f ! . f(2), f(4), f(6) , , -

, x0! (2,6) , !!f (x

0)= 0 .

9. f(x)= !2x6 + "x4 + x2 + #+ $ , ! ,",#,$! !* , 3"2 < 5!2 . , ' -

.

10.

f

[!

,"

],

f(!) 0 .

Bolzano, x0! (!,") , g(x

0)= 0 .

") :

. f [!,x0] , [x

0,"] .

. f (!,x0) , (x

0,") .

, :

!1! (",x

0) , !f(!

1)=

f(x0)"f(")

x0""

=

(") "+ #"x0""

x0""

# !f(!1)=

#"x0

x0""

.

!2! (x

0,") , !f(!

2)=

f(")"f(x0)

""x0

=

(#) ""#""+ x0

""x0

# !f(!2)=

x0"#

""x0

.

!f(!1) " !f(!

2)=

"#x0

x0##

"x0##

"#x0

$ !f(!1) " !f(!

2)= 1 .

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2. f: [!,"]! ! , [!,"] (!,") ,

f(!)= 2" f(!)= 2" .

) f(x)= 2x, , (!,") .

) !1 ,!2 ! (",#) , !

f(!1) " !f(!2)= 4 .

!) f(x)!2x= 0 .

g(x)=f(x)!2x, x" [!,"] , g(x)= 0

, , (!,") .

. g [!,"] .

. g(!)=f(!)!2! = 2"!2!" g(!)= 2("!!)> 0 ,

[!,"] !>

" .

. g(!)=f(!)!2!= 2"!2!" g(!)=!2(!!")< 0 .

Bolzano, g(x)= 0!f(x)= 2x,

, (!,") , x0! (!,") , f(x

0)= 2x

0(1)

") :

. f [!,x0] , [x

0,"] .

. f (!,x0) , (x

0,") .

, :

!1! (",x

0) , !f(!

1)=

f(x0)"f(")

x0""

=

(1) 2x0"2#

x0""

# !f(!1)=

2(x0"#)

x0""

.

!2! (x

0,") , !f(!

2)=

f(")"f(x0)

""x0

=

(1) 2#"2x0

""x0

=

"2(x0"#)

"(x0"")

#

! "f(!2)

=

2(x0#")

x0## .

!f(!1) " !f(!

2)=

2(x0# ")

x0##

"2(x

0##)

x0# "

$ !f(!1) " !f(!

2)=4 .

!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.

!"#$%$&7.!"#$%&'()%*+,-*(!'+).


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3. f, [1,3] .

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

) [1,2] [2,3] .

)

x0!

(1,3) , !!

f (x0)=

0 .

!) f [1,3] :

!f , f () [1,3] .

!!f , !f () [1,3] .

, [1,2] , [2,3] , :

x1! (1,2) , !f(x

1)=

f(2)"f(1)

2"1# !f(x

1)=f(2)"f(1) .

x2! (2,3) , !f(x

2)=f(3)"f(2)

3"2# !f(x

2)=f(3)"f(2) .

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") :

. !f [x1,x

2] .

. !f (x1,x

2) .

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

f(2)+f(2)=f(1)+f(3)!f(2)"f(1)=f(3)"f(2)!(!)

#f(x1)= #f(x

2) .

Rolle, x0! (x

1,x

2) , (1,3) , -

!!f (x0)= 0 .

!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.

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4. f:[0,1]! ! , f(0)= 0 ,f(1)= 1 . -

:

) x0! (0,1) , f(x

0)=

1

2.

) !1,!

2! (0,1) , 1

!f(!1)+ 1

!f(!2)= 2 .

!) x0= x

f(x)=

1

2! 2f(x)= 1! 2f(x)"1= 0 .

g(x)= 2f(x)!1 , x" [0,1] , x0! (0,1) ,

g(x0)= 0 .

. g [0,1] -

([0,1] , f ' ).

. g(0)= 2f(0)!1= 2 "0!1# g(0)=!1< 0 .

III. g(1)= 2f(1)!1= 2 "1!1# g(1)= 1> 0 .

Bolzano, x0! (0,1) , g(x

0)= 0 .

") :

. f [0,x0] , [x

0,1] .

. f (0,x0) , (x

0,1) .

, :

!1! (0,x

0) , !f(!

1)=

f(x0)"f(0)

x0"0

=

(")1

2"0

x0

# !f(!1)=

1

2x0

.

!2 ! (x0 ,1) , !f(!2)=f(1)"f(x0)

1"x0

=

(") 1"1

21"x

0

=

1

21"x

0

# !f(!2)= 12(1"x0).

1

!f(!1)+

1

!f(!2)= 2x

0+ 2(1"x

0)= 2x

0+1"2x

0#

1

!f(!1)+

1

!f(!2)= 2 .

!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.

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6. f: [!,"]! ! , f(!)> 0 ,f(")> 0 ,

f(!)< 0 , !! (",#) . :

) f(x)= 0 , , (!,") .

) x0! (!,") , C

fx

0, -

x'x.

) !! (",#) , !!f (!)> 0 .

!) f [!,"] :

!f , f () [!,"] .

!!f , !f () [!,"] .

:

. f [!,"] , [",#] .

. f(!) !f(")< 0 , f(") !f(#)< 0 .

Bolzano, f(x)= 0 , ,

x1! (!,") , , x

2! (!,") , , ,

(!,") .

") x0 ! (!,") , !

f(x0)=

0 .

. f [x1,x

2] .

. f (x1,x

2) .

. f(x1)=f(x

2)= 0 , x

1,x

2f(x)= 0 .

Rolle, x0! (x

1,x

2) , (!,") , -

!f(x0)= 0 .

#) :

. f [!,x1] , [x

2,"] .

. f (!,x1) , (x

2,") .

, :

!1! (",x

1) , !f(!

1)=

f(x1)"f(")

x1""

="f(")

x1""

< 0 ,

f(!)> 0 x1!!> 0 .

!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.

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!2! (x

2,") , !f(!

2)=

f(")"f(x2)

"" x2

=f(")

"" x2

> 0 ,

f(!)> 0 !!x2> 0 .

:

. !f [!1,!

2] .

. !f (!1,!

2) .

, !! (!1,!

2) , (!,") ,

!!f (!)=!f(!

2)" !f(!

1)

!2" !

1

.

, !f(!2)> 0 , !f(!

1)< 0 , !

2"!

1> 0 , !!f (!)> 0 .

7. f ! -

Cf

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

!(!,f(!)) , "(",f(")) , #(#,f(#)) , !< "< # ,

Cf

.

!"! "!!

// "#! "!

$ det(!"! "!!

,"#! "!

)= 0 (1)

!"

! "!!

= (x"#x! ,y"#y!)= (!#" ,f(!)#f(")) .

!"! "!

= (x"#x

!,y

"#y

!)= (!#" ,f(!)#f(")) .

det(!"! "!!

,"#! "!

)=!$" f(!)$f(")

#$! f(#)$f(!)=(!$") %[f(#)$f(!)]$(#$!)%[f(!)$f(")] .

(1) (!!")"[f(#)!f(!)]!(#!!) "[f(!)!f(")]= 0#

! (!"")#[f(#)"f(!)]= (#"!) #[f(!)"f(")] .

!< "< # !!"> 0 , #

!!> 0 ,

f(!)!f(")

!!"=

f(#)!f(!)

#!!(2)

f ! :

!f , f () ! .

!!f , !f () ! .

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:

. f [!,"] , [",#] .

. f (!,") , (",#) .

,

x

1! (!,") , !f(x

1)=

f(!)"f(")

!"", x

2! (!,") , !f(x

2)=

f(!)"f(")

!" ".

(2) , !f(x1)= !f(x

2) .

, !f [x1,x

2] (x

1,x

2) ,

Rolle, !! (x1,x

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

8. f ! . f(2), f(4), f(6)

, , -

, x0! (2,6) , !!f (x

0)= 0 .

f(2), f(4), f(6) ,

f(2)+f(6)= 2f(4)!f(2)+f(6)=f(4)+f(4)!f(4)"f(2)=f(6)"f(4)!

!

f(4)"f(2)

4"2=

f(6)"f(4)

6"4(1)

f ! :

!f , f () ! .

!!f , !f () ! .

:

. f [2,4] , [4,6] .

. f (2,4) , (4,6) .

,

x

1! (2,4) , !f(x

1)=

f(4)"f(2)

4 " 2, x

2! (4,6) , !f(x

2)=

f(6)"f(4)

6"4.

(2) , !f(x1)= !f(x

2) .

, !f [x1,x

2] (x

1,x

2) ,

Rolle, x0! (x

1,x

2) , (2,6) ,

!!f (x0)= 0 .

!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.

!"#$%$&7.!"#$%&'()%*+,-*(!'+).


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9. f(x)= !2x6 + "x4 + x2 + #+ $ , ! ,",#,$! !* , 3"2 < 5!2 . , ' -

.

!(x1

,f(x1

)) , "(x2

,f(x2

)) , #(x3

,f(x3

)) , x1< x

2< x

3, -

f.

!"! "!!

// "#! "!

$ det(!"! "!!

,"#! "!

)= 0 (1)

!"! "!!

= (x"#x

!,y

"#y

!)= (x

2#x

1,f(x

2)#f(x

1)) .

!"! "!

= (x"#x

!,y

"#y

!)= (x

3#x

2,f(x

3)#f(x

2)) .

det(!"! "!!

,"#! "!

)=x

2$x

1 f(x

2)$f(x

1)

x3

$x2

f(x3

)$f(x2

)=(x

2$x

1) %[f(x

3)$f(x

2)]$(x

3$x

2) %[f(x

2)$f(x

1)] .

(1) (x2!x

1) "[f(x

3)!f(x

2)]!(x

3!x

2) "[f(x

2)!f(x

1)]=0#

! (x

2"x

1) #[f(x

3)"f(x

2)]=(x

3"x

2) #[f(x

2)"f(x

1)] .

x1< x

2< x

3x

2!x

1> 0 , x

3!x

2> 0 ,

f(x2)!f(x

1)

x2!x

1

=

f(x3)!f(x

2)

x3!x

2

(2)

f ! :

!f , f () ! .

!!f , !f () ! .

:

. f [x1,x

2] , [x

2,x

3] .

. f (x1,x

2) , (x

2,x

3) .

,

!1! (x

1,x

2) , !f(!

1)=

f(x2)"f(x

1)

x2"x

1

, !2! (x

2,x

3) , !f(!

2)=

f(x3)"f(x

2)

x3"x

2

.

(2) , !f(!1)= !f(!

2) .

!f(x)= 6!2x5 + 4"x3 + 2x, [!1,!

2] (!

1,!

2)

.

!f(!1)= !f(!

2) , Rolle

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

!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.

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!!f (x)= 0 30!2x4 +12"x2 + 2= 0!15!2x4 + 6"x2 +1= 0 .

x2 = t! 0 15!2t2 + 6"t+1= 0 ,

!= (6!)2" 4 #15"2 #1= 36!2"60"2 = 12(3!2 "5"2)< 0 , 3!2 < 5"2 .

!!f (x)= 0 , .

, ,

' Cf

.

10. f [!,"] , f(!)< 0

f(!)= !f(!)= 0 . !! (",#) , !!f (!)< 0 .

f [!,"] :

!f , f () [!,"] .

!!f , !f () [!,"] .

f [!,"] (!,") ,

x0! (!,") , !f(x

0)=

f(!)"f(")

!""# !f(x

0)=

f(!)

!"".

f(!)< 0 !!"> 0 ([!,"] !> " ),

!

f(x0)< 0 .

!f [!,x0] (!,x

0) , -

!! (",x0) , (!,") ,

!!f (!)=!f(x

0)" !f(")

x0""

# !!f (!)=!f(x

0)

x0""

.

!f(x0)< 0 , , x

0!!> 0 , '

(!,x0) .

, !! (",#) , !!

f (!)< 0 .

!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.

!"#$%$&7.!"#$%&'()%*+,-*(!'+).


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