-
7/25/2019 43- (5)
1/14
!"#$%"&'($. )*+,-.*/0*&
!"#$%$&'()"#$"&*+,-
.&/0*1201(34*5&1&67681/."-69*5&$81/:655"71-8+$(
!"#$%"&'()
*+,-(./0-
!"#$%&'#()*- +%,()%-2015 - 2."/'%0.
!"#$%&'()*%+,-&.+(!(,)
10!"!#$%&'(#$)"*+!,'-,*&+
!"#$%&'(" 43
!"#$"%&'()*+ %&)-&.(+ /. '%0.12-/%'% '1#
Bolzano,
Rolle)%( '3 456.
-
7/25/2019 43- (5)
2/14
!"#$%&'()*+,-.,%('/#0/12%1 - 6)
12 + 1 30'('/(,4!2'5,62#*7,82/*, 915".'2,2,- :;7-#1
-
7/25/2019 43- (5)
3/14
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 .
-
7/25/2019 43- (5)
4/14
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 .
!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.
!"#$%$&7.!"#$%&'()%*+,-*(!'+).
)23"2$51-+$715- www.mathsteki.gr
- 4 -
-
7/25/2019 43- (5)
6/14
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$516(.
!"#$%$&7.!"#$%&'()%*+,-*(!'+).
)23"2$51-+$715- www.mathsteki.gr
- 5 -
-
7/25/2019 43- (5)
7/14
3. f, [1,3] .
2f(2)=f(1)+f(3) :
) [1,2] [2,3] .
)
x0!
(1,3) , !!
f (x0)=
0 .
!) f [1,3] :
!f , f () [1,3] .
!!f , !f () [1,3] .
, [1,2] , [2,3] , :
x1! (1,2) , !f(x
1)=
f(2)"f(1)
2"1# !f(x
1)=f(2)"f(1) .
x2! (2,3) , !f(x
2)=f(3)"f(2)
3"2# !f(x
2)=f(3)"f(2) .
!"#$%&. !"#$"%&'#(#()*+,'()-./'(0"[", 1]
'0"23#.3"#4.1#1".5'#(#&(..'67)83).4%)94):;'#.%/?#(#()'84;%"'"4)84%)=74(#..
") :
. !f [x1,x
2] .
. !f (x1,x
2) .
. 2f(2)=f(1)+f(3)
f(2)+f(2)=f(1)+f(3)!f(2)"f(1)=f(3)"f(2)!(!)
#f(x1)= #f(x
2) .
Rolle, x0! (x
1,x
2) , (1,3) , -
!!f (x0)= 0 .
!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.
'()*+*,7.*#5%0"+;'0
-
7/25/2019 43- (5)
8/14
4. f:[0,1]! ! , f(0)= 0 ,f(1)= 1 . -
:
) x0! (0,1) , f(x
0)=
1
2.
) !1,!
2! (0,1) , 1
!f(!1)+ 1
!f(!2)= 2 .
!) x0= x
f(x)=
1
2! 2f(x)= 1! 2f(x)"1= 0 .
g(x)= 2f(x)!1 , x" [0,1] , x0! (0,1) ,
g(x0)= 0 .
. g [0,1] -
([0,1] , f ' ).
. g(0)= 2f(0)!1= 2 "0!1# g(0)=!1< 0 .
III. g(1)= 2f(1)!1= 2 "1!1# g(1)= 1> 0 .
Bolzano, x0! (0,1) , g(x
0)= 0 .
") :
. f [0,x0] , [x
0,1] .
. f (0,x0) , (x
0,1) .
, :
!1! (0,x
0) , !f(!
1)=
f(x0)"f(0)
x0"0
=
(")1
2"0
x0
# !f(!1)=
1
2x0
.
!2 ! (x0 ,1) , !f(!2)=f(1)"f(x0)
1"x0
=
(") 1"1
21"x
0
=
1
21"x
0
# !f(!2)= 12(1"x0).
1
!f(!1)+
1
!f(!2)= 2x
0+ 2(1"x
0)= 2x
0+1"2x
0#
1
!f(!1)+
1
!f(!2)= 2 .
!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.
!"#$%$&7.!"#$%&'()%*+,-*(!'+).
)23"2$51-+$715- www.mathsteki.gr
- 7 -
-
7/25/2019 43- (5)
9/14
-
7/25/2019 43- (5)
10/14
6. f: [!,"]! ! , f(!)> 0 ,f(")> 0 ,
f(!)< 0 , !! (",#) . :
) f(x)= 0 , , (!,") .
) x0! (!,") , C
fx
0, -
x'x.
) !! (",#) , !!f (!)> 0 .
!) f [!,"] :
!f , f () [!,"] .
!!f , !f () [!,"] .
:
. f [!,"] , [",#] .
. f(!) !f(")< 0 , f(") !f(#)< 0 .
Bolzano, f(x)= 0 , ,
x1! (!,") , , x
2! (!,") , , ,
(!,") .
") x0 ! (!,") , !
f(x0)=
0 .
. f [x1,x
2] .
. f (x1,x
2) .
. f(x1)=f(x
2)= 0 , x
1,x
2f(x)= 0 .
Rolle, x0! (x
1,x
2) , (!,") , -
!f(x0)= 0 .
#) :
. f [!,x1] , [x
2,"] .
. f (!,x1) , (x
2,") .
, :
!1! (",x
1) , !f(!
1)=
f(x1)"f(")
x1""
="f(")
x1""
< 0 ,
f(!)> 0 x1!!> 0 .
!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.
!"#$%$&7.!"#$%&'()%*+,-*(!'+).
)23"2$51-+$715- www.mathsteki.gr
- 9 -
-
7/25/2019 43- (5)
11/14
!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(!)$f(")] .
(1) (!!")"[f(#)!f(!)]!(#!!) "[f(!)!f(")]= 0#
! (!"")#[f(#)"f(!)]= (#"!) #[f(!)"f(")] .
!< "< # !!"> 0 , #
!!> 0 ,
f(!)!f(")
!!"=
f(#)!f(!)
#!!(2)
f ! :
!f , f () ! .
!!f , !f () ! .
!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.
!"#$%$&7.!"#$%&'()%*+,-*(!'+).
)23"2$51-+$715- www.mathsteki.gr
- 10 -
-
7/25/2019 43- (5)
12/14
:
. f [!,"] , [",#] .
. f (!,") , (",#) .
,
x
1! (!,") , !f(x
1)=
f(!)"f(")
!"", x
2! (!,") , !f(x
2)=
f(!)"f(")
!" ".
(2) , !f(x1)= !f(x
2) .
, !f [x1,x
2] (x
1,x
2) ,
Rolle, !! (x1,x
2) , !!f (!)= 0 .
8. f ! . f(2), f(4), f(6)
, , -
, x0! (2,6) , !!f (x
0)= 0 .
f(2), f(4), f(6) ,
f(2)+f(6)= 2f(4)!f(2)+f(6)=f(4)+f(4)!f(4)"f(2)=f(6)"f(4)!
!
f(4)"f(2)
4"2=
f(6)"f(4)
6"4(1)
f ! :
!f , f () ! .
!!f , !f () ! .
:
. f [2,4] , [4,6] .
. f (2,4) , (4,6) .
,
x
1! (2,4) , !f(x
1)=
f(4)"f(2)
4 " 2, x
2! (4,6) , !f(x
2)=
f(6)"f(4)
6"4.
(2) , !f(x1)= !f(x
2) .
, !f [x1,x
2] (x
1,x
2) ,
Rolle, x0! (x
1,x
2) , (2,6) ,
!!f (x0)= 0 .
!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.
!"#$%$&7.!"#$%&'()%*+,-*(!'+).
)23"2$51-+$715- www.mathsteki.gr
- 11 -
-
7/25/2019 43- (5)
13/14
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$516(.
!"#$%$&7.!"#$%&'()%*+,-*(!'+).
)23"2$51-+$715- www.mathsteki.gr
- 12 -
-
7/25/2019 43- (5)
14/14
!!f (x)= 0 30!2x4 +12"x2 + 2= 0!15!2x4 + 6"x2 +1= 0 .
x2 = t! 0 15!2t2 + 6"t+1= 0 ,
!= (6!)2" 4 #15"2 #1= 36!2"60"2 = 12(3!2 "5"2)< 0 , 3!2 <
5"2 .
!!f (x)= 0 , .
, ,
' Cf
.
10. f [!,"] , f(!)< 0
f(!)= !f(!)= 0 . !! (",#) , !!f (!)< 0 .
f [!,"] :
!f , f () [!,"] .
!!f , !f () [!,"] .
f [!,"] (!,") ,
x0! (!,") , !f(x
0)=
f(!)"f(")
!""# !f(x
0)=
f(!)
!"".
f(!)< 0 !!"> 0 ([!,"] !> " ),
!
f(x0)< 0 .
!f [!,x0] (!,x
0) , -
!! (",x0) , (!,") ,
!!f (!)=!f(x
0)" !f(")
x0""
# !!f (!)=!f(x
0)
x0""
.
!f(x0)< 0 , , x
0!!> 0 , '
(!,x0) .
, !! (",#) , !!
f (!)< 0 .
!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.
!"#$%$&7.!"#$%&'()%*+,-*(!'+).
)23"2$51-+$715- www.mathsteki.gr
13
