Sáng kiến kinh nghiệm: Chứng minh bất đẳng thức bằng phương pháp đạo hàm

ng dng đo hm đ chng minh bt đng thc
Đ 
!"#$%&'()!!"$%
!*$+ +,-!./&)$%!*$'
012+&!3+#
),4564#+.7++89%!*$8"-
#89)!'0,8-!9,!
##8:#!9!9;

4!9,&!.9)<+8
.:)=>8:!&<+.7?!".@!?'
AB)>!C9&"DE-F'G%
<!./?+HIJ!3+!"!9!./7'
Trang 1
ng dng đo hm đ chng minh bt đng thc
K$)!!"$89%!*$8%
!97L!I,M!B-N-O8&.-/+H
!?-#&,'P?8?-#!)=)
2!-%!*$M$'
4>,Q%JR?"-
/+  8:  9  %  !*  $      .;  S%  !*  $  T,
S+4#. +UU-/+8:$1,+'
Sau đây l mt s bi ton v bt đng thc dng phương php trên đ# gi$i:
Bi 1:T&DC; 'T$1; '
Hư'ng d(n:VB 'K!
W &;
4;
SS4;
X
Y Z[


\
]<, SV4!./$'
T*ng qut hơn:X^T&DC;[_'T$%!*
$;
 '
\^T&DC
Trang 2
ng dng đo hm đ chng minh bt đng thc
T$; '
Bi 2: T&I'T$1;
Hư'ng d(n:4%!*$;
Y 0-_ZRNXO!28:
Y 0-`ZR
VB
SS4;
X
Y Z[

X
]<, SV4!./$'
Bi 3:T T$1; '
Hư'ng d(n:]: ;
TM$; ,
W &
4 !3-#
Trang 3
ng dng đo hm đ chng minh bt đng thc
P!8:  SV4!./$'
Bi 4:T$1;0-F`Z&,#).7R;
Hư'ng d(n:VB '

TM$
Y4;
Yab 4$
4<8<,;  &   !3-# 
'P! 
]<, SV4!./$'
Bi 5:T c$1;
Hư'ng d(n:SV4NXO
W & '4;
W     &   '  4  ;  
& d-#
e, ,& d-# '
Trang 4
ng dng đo hm đ chng minh bt đng thc
4J!-b R , '
f+)SV44#. +\)C,&; 8N SV4M
$'+'
Bi 6:T$1;0-+.7R 
R 'P%!*$F,g
Hư'ng d(n:ab+.7RF
Z
R 8
VB !./+.7R;
'P!;
W &; 8: 
'4SS4;

[

]<, SV4 ')%!*$F,;

Trang 5
ng dng đo hm đ chng minh bt đng thc

Bi 7:T$1;0- R
Hư'ng d(n:W &; 8
]: R   ,  )%h_iF, 
'
' e,;  )%h_iF,
 '
]<, 8:
Bi 8:a]e"8)FRjF,'
T$1;
Hư'ng d(n:4; N !,k !.@
(1)
VB F &; 
Trang 6
ng dng đo hm đ chng minh bt đng thc
'4SS4;
 
[ZY

]<,   SV4!./$'
Bi 9:T DC
T$1;
Hư'ng d(n:4J-,;
N
W &; 8:
4.7?l;
mM./, 8N\O38-Q8-!./SV4NXO'
Bi 10:T$1; P%!*$F,

g+'
Hư'ng d(n:SV4!C NXO
Trang 7
ng dng đo hm đ chng minh bt đng thc
W &;
VB
0- R  R
VB J 8!./&
& !3-#
SV4!./$'
P%!*$F, ,
Bi 11:T T$1; NXO
Hư'ng d(n:SV4NXO N\O
VB ) # 'SV4N\O
• T$; '
VB d-#
P!8: R 
• T$;
VB   'T$.7?!./   !3-# 
Trang 8
ng dng đo hm đ chng minh bt đng thc
,
4J!,SV4M!./$'
Bi 12:T$1;]: R 'NXO
Hư'ng d(n:VB 8: '
4;NXO
  
[ZYZ[
]<, SV4M$'
Bi 13:T T$1; !9;

Hư'ng d(n:W &;
4;
]: R & !3-#
e,  & !3-#
P!
]<, SV4!./$'
Bi 14:T$1;
Trang 9
ng dng đo hm đ chng minh bt đng thc
f+  )  $    1;  0-  \  &   D  C  NXO  R; 
Hư'ng d(n:W &;
4;  SS4;
F
 
YZ[


e,    SV4  NXO
!./$'
f+);n0- RN\ODC
0- RN\O '
VB R SV4N\O!./$'
Bi 15:Tc& 'T$1;+'
[
Hư'ng d(n:VB 'W &;
[
Trang 10
ng dng đo hm đ chng minh bt đng thc
4; !
& !3-#

4F c.@/+;
• 45X;  4;
• 45\;  4;
 
• 45c; )%,!o# '4SS4;
 
YZ[
e,;
A # ']<, 

Trang 11
ng dng đo hm đ chng minh bt đng thc
Bi 1:T$1;]: %!*$;
N5P;W &; 8: '
N5P;W &; 8: '
Bi 2:T c$1;
HD:W &; 8:
Bi 3:T T$1; '
Bi 4:T 'T$1; '
Trang 12
ng dng đo hm đ chng minh bt đng thc
HD:W &; 8: 8$ d-#
Bi 5:T T$1;
HD:VB 'W &;
Bi 6:T$1;0-  R
Bi 7:]: 'T$1; [ 8: '

Trang 13
ng dng đo hm đ chng minh bt đng thc
Trang 14