BÀI TẬP CHỌN LỌC VỀ PHƯƠNG PHÁP HÀM SỐ
Bài 1.

m







− !
"

#
mxxx =++−− "" $
"$"
%
Rm
∈
&#
Giải
'(
( )
2 3 2
3 1 2 2 1f x x x x= − − + +
)
( )
f x
*+,-./.
!

∀ ∈ −
 
 

2 3 2
4 3 3 4
3 4 0 0
3
1 2 1
x
x x
x x x
+
> − ⇒ + > ⇒ + >
− + +
#
123
( )
0 0 0f x x= ⇔ =
#
4567.3
( )
( )
0 88 88
1
0 1
2
0
1
CÑ

( )
"
 * "* " *%" *& = %"&− + + + − ≤
*
=! $
 
∈ +
 
Giải
'(
2
t x 2x 2= − +
⇔
"
−">*
"
−"*
4%"&⇔
−
≤ ≤ ≤ ∈ +
+
2
t 2
m (1 t 2),dox [0;1 3]
t 1
?5
2
t 2
g(t)
t 1

≤ = =
t 1;2
2
m maxg(t) g(2)
3
12
≤
"
$
#
Bài 3.
m:3

"
" "% D& E = $ =x m x m x− + + + − + =
#
Giải
"
" "% D& E = $ =x m x m x− + + + − + =
"
" "% D& E = $x m x m x⇔ − + + + = −

" "
$ =
" "% D& E = % $&
x
x m x m x
− ≥



x
− +
=
−
"
"
"% E &
0% &
%" E&
x x
f x
x
−
⇒ =
−
?3
4567.3
x H
∞
= EI" $E J
∞
K J = H H = J

L
"DIE
J
∞
; 
"D
%L! &

= + − = + −
 
( ) ( ) ( )
2
2
1
2t t 1 4t 2t t 1 t 1
2
 
= + − + − + −
 
=
( )
( )
2
1
3t 1 7t 4t 1
2
− − +
B¶ng biÕn thiªn
Qua b¶ng biÕn thiªn ta cã : miny =
1
27
vµ maxy = 3.
Bài 5.
N
3 3
cos sinx x m− =
% &
& O5P>H 

π π

= +

∈

= +


¢
6&% &
( ) ( )
cos sin 1 cos sinx x x x m⇔ − + =

'(>
cos sinx x
−
!SP
2t ≤
#
?
; 0; 2
4 4
x t
π π
 
 
∈ − ⇒ ∈
 
 

W%&
H I$
J
=
H
$
1
27

Bi 6.
m3
$ $ "
" " "
$ $ " =% &
$ " =%"&
x y y x
x x y y m

+ =


+ + =


:
Gii
'SP3
"
"
=

% & >yy>xJ %"&
" "
" =x x m + =
'(
"
v x=
vB=! C%"&v
"
J"v >m#
X,g%v&>v
"
J"v
=! =!
% & ! % & "
[ ] [ ]
axg v g v= =
12P+,VP m"#
Bi 7.
N:PYZ
zyx ))
5<
$
"""
=++ zyx
#-@6[
zyx
zxyzxyA
++
+++=
E

$
"
t
t
A +

=
Xét hàm số
#$$)
"
$E
"
&%
"
+= t
t
t
tf
Ta có
=
EE
&%0
"
$
"
>

==
t
t

"
J$H>=
2
2
2 3 3
1
t t
m
t t
+
=
+
%"&
FG,W%&>
2
2
2 3 3
1
t t
t t
+
+
%=&
^26567.
% & %"&=
5
3
3
m
#

#
[ ] [ ]
=! " =! =! #
"
x x t



_
( )
[ ]
"
$ " $ ) =! f t t t m t= + + =
6567.
`<.
=
=! "
" $
m


#
Bi 10.
Cho hai số thực x, y thoả mãn :




+ + + =


Đặt
! " % =! =&u x v y u v= + = +
Ta có hệ phơng trình:
( )
( )
"
" "
"
" $
$$

$
$
$
" \
a
u v
u v uv a
u v a
a
a
u v a
u v
uv a
+ =
ữ
ữ (*)
Hệ (I) có nghiệm khi và chỉ khi phơng trình (*) có hai nghiệm t
1
, t
2
không âm
"
"
= L ED =
\ $ "
= = \ $ E
"
=
\ "a =
a a
S a a
P
a a

+ +

+


" $ =
x x
x m x m
+
+ +





Gii
X6
( )
"
"
a b = % &
" $ = %"&
x x
x m x m

+


+ +


( )
bx
#X<P+,VPc
[ ]

X<
[ ]
= =
!b 3 % &x f x m⇔ ∃ ∈ ≥
( )
( )
( )
( )
"
"
" "
" D
" " L
0
" "
x x
x x
f x
x x
+ −
+ −
= =
+ +
!
( )
"
a
0 = D =
"
f x x x x

= =
!b
"a
!b 3 % & * % &
$
x
x f x m f x m m
∈
∃ ∈ ≥ ⇔ ≥ ⇔ ≥
#
Bài 12.
O5
(
)
" "
"
$ $
$
x y x x y
x y x x

+ + + = −



+ + = +

Giải
'P
"

#
Giải
;
( ) ( )
$
D
" " x x m x x x x m+ − + − − − =
% &
'SP3
= x
≤ ≤
g7 
[ ]
=! x ∈
 M<% & ]*hM<% &.% &eSP"
x x x= − ⇒ =
#

"
x =
+,% &f3
$
=

"# "#

" "

" " = =
x x x x x x
x x x x x x x x x x x x
+ − − − − − = −
⇔ + − − − + + − − − = ⇔ − − + − − =
J1@
D D

=
"
x x x− − = ⇔ =
J1@

=
"
x x x− − = ⇔ =
jf,)% &h#
E
i1@> % &R,3
( ) ( )
( ) ( )
" "
D D
D
" " x x x x x x x x x x+ − − − = − − ⇔ − − = − −
% &"

=)
"
x x= =

% E & #E #%E & ### #%E &x C C x C x C x− = − + − +
gZ"+7+@*f3
$= = " " " $ $= $= $
$= $= $= $=
% E & #E #E ### #Ex x C x C x C x C x− = − + − +
% &
^,+7% &f!
$= "\ = " " " $= $= $=
$= $= $= $=
% E & E= % E & " #E $ #E ### $ #Ex x x C C x C x C x− − − = − + − +
%"&
Nd*>H +,%"&f
$= "\ = " " $= $=
$= $= $= $=
b E=#b "% #E& $% #E & ### $ % #E &C C C C+ = + + + +

"\
= " $=
b %b E=& " $ ### $ a a a a+ = + + + +

$=
= " $=
b #"b " $ ### $ a a a a= + + + +
Đ_3
$=
b #"bS =
.
Bài 15.
"Z63


x
t
x
'SP3–2< t
E≤
#
 mQm3m=
"
" "+t
t
#^2656..⇒
"
D
E
< ≤m
hoặc –5 <
D< −m
Bài 16.
O5+,6-23
mx m x mx x x x
2 2 3 2
1.( 2 2 ) 3 4 2+ + + = − + −
%D&
Giải
%D&⇔
$ $
% & % & % &+ + + = − + −mx mx x x
#
FG,3W%&>
$

P x y
xy
+ −
= + +
#
Giải

" "
$x y xy x y xy+ = + +

% & $ % &xy x y x y xy⇔ + = + +
*p=!p=.*Jp=
% &
( )
"
D
$ $ $% & D =x y x y x y
x y x y
⇒ + = + + ≥ + ⇒ + − + − ≥
+
b
( )
[ ]
% & D = Dx y x y x y⇒  + +  + − ≥ ⇒ + ≥
 
q,  ;  >  %*  J  &
"
 J  "  H 

xy

$ " $
= pD
t
t t
−
= > ∀
,
% &f t
-./.rP5
[
)
D!+∞
g.
% &f t
c67.rP5
[
)
D!+∞
>p
a
% & %D&
D
P f t f= ≥ =
#XM;6U
a
D
P*>>"#
Bài 18.
63
$ "

*
H
$− +

+∞
K J= H


$ " $
$
+

= =
12
a $
=!
$
t
 
+
 
∈
 
 
#4<P+,VP% &
a $
=!
$
t
 


D
`6567.3
a $
B=! C
$

* % &
D "
t
M f t khi x
+
∈
= =
126+@

D
m ≤
#
Bài 19.
Na)b)c-,:PYZM<
a b c+ + =
#N[U3
a
"
"a
ab bc ca abc+ + − ≤
#
Giải
a

% & a a
%" &
D "a D $ $ "a
a
f a a
 
−
 
= − + − ≤
 ÷
 ÷
 ÷
 
 
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Bài 20.
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