Vận dụng hàm số vào việc giải phương trình, bất phương trình và hệ phương trình SKKN toán 12

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Ng*+, /0.,12 Taàn Theá Anh
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&'!()%*!
3.42..5267642.82.,129:-;.<0

Naêm hoïc 2011 - 2012
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=>=?=@$A
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BG9HI252.J$2.
BKHL <2K2M6(,2.+, /-/01.
NB,O,-P2.23!
QB!RS0.T#45#6#789:;
UB!,12 VW,.0/1<.=,>/
XBYSZ[[BB\6S,]-S2 \S2.^U^_`K6S,]B0V6
aB.b09c
_B!d29R032K-<0#45#6#789:;B
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M62.:2no2K+ +

.hLI22K.H2.jHV-WV#
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=
eD iS2K
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6B2CC2KCEL2MGNO7I#JC
#;4;4P%(Q;8,
O#RS8%T%A,
3#RSU
9T%AU
+7V;SU
>!W3(Q;8<
,<
6B2CCC#XG6YG#6ZG6C[27I#JC<
CG\]R<
CC2 (Q;8=
O$;:;=
C#)((^_;4`8!\V_;4`Pa;_;4`
ba;W38!\V =
CC8!\V^R;4c_a;
;4c=
dd8;R1
CY(e;%"3;3!\V1
CCY(e;%"3;3!\V/.
Gd8;R(Q`/>

6B2Cf9g#Lh7i#7jkG&OL96C#6ZG6C[27I#JC/,
6B2Cf9g#$Ll2/U
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( )
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=

B2^!W3a;;4cuuxw≥vuxw8
‚8( ;P‚
(~;`
( )
y u x
=
ƒ!])3;4
\P‚(~;`
( )
y v x
=

NB2^!W3a;;4cuuxw≤vuxw8
‚8( ;P‚(~;`
( )
y u x
=
ƒ!])3P\P‚(~;`
( )
y v x
=

QB2^!W3;4cuuxw=m88( 
3(e!W3(5;ry=mP(~;`
( )
y u x
=


UBd#uuxw≥m^!({∀x∈C⇔
( )
C
m
x
u x m
∈
≥
XBd#uuxw≤m ({∀x∈C⇔
( )
C
m3t
x
u x m
∈
≤
aBd#uuxw≥mT^!x∈C⇔
( )
C
m3t
x
u x m
∈
≥
_Bd#uuxw≤mT^!x∈C⇔
( )
C
m
x
u x m

∈
≤
.„…Hàm tăng giảm nghiêm ngặt.
Mệnh đề 1: Xét phương trình f(x) = m, m là hằng số
x D
∈
. Nếu trên miền D hàm số
f(x) đồng biến ( Hoặc nghịch biến) và phương trình có nghiệm thì nghiệm đó là duy
nhất.
Mệnh đề 2: Xét phương trình f(x) = g(x) với
x D
∈
. Nếu trên miền D hàm f(x) đồng
biến và g(x) nghịch biến và nếu phương trình có nghiệm thì nghiệm đó là duy nhất.
B}
SB}C}$$
H,.V|.*d2K-i42.
U +
+ / .x x x
− − − =
u/wG!;4cT
^!a;u7Q%VE+ ,w
,8,
#3T
U +
u /wx x
= +
u+w_;„u+w
+
. t . utŒ/w /x khi⇒ ≥ ≥ ⇒ ≥

_R•;„u+w;3T
U
/ /x x
≥ ⇒ ≥

2R!^!W3;4cu/wu:Tw;c
/x
≥
2
U +
u w + / .
u/w
/
f x x x x
x

= − − − =
⇔

≥

u>w
#3T
, , , ,
Žu w U + + u+ + w u+ +w . /f x x x x x x x x= − − = − + − + > ∀ ≥
mp;%…utw;
/x
∀ ≥
_\438!\V…utw(~:
/x

∀ ≥
u•w
m8…u/w…u+w•.u+•w
α
β
b
x
a
v(x)
u(x)
a
b
x
y = m


#„u•w8u+•w\43;4cu/wT^!a;
2Rt‘;7Qx3;4%888\Rt‘;(S
;4cT^!;c
/≥x
8:;R(`343
H,,8,.1|.*d2K-i42.
>
/ /
+ /
x y
x y
y x

− = −




= +

u/wu9VO+ >w M
fP(%
 .x y
≠
_;3T
>
>
>
u+w
+ / .
/
u wu/ w .
u/w
/
u/ w .
+ /
u>w
+ /
x y
x x
x y
xy
y x
xy
y x


=



− + =



− + =


⇔ ⇔



+ =


= +





= +


Mu+w
/ U / U

u ‹ w ’u/‹/w‹u ‹ w“
+ +
x y
− ± − ±
=
Mu>w_
,
/
+ .
y
x
x x
−

=



+ + =

”‘;8!\V…utw†
,
+x x
+ +
P
.x
≠
m…utw‡.
.x
∀ ≠

_^;4cu>w"^!
Chú ý: Rất nhiều học sinh giải bài toán theo hướng :
7p;
+
/ /
u w Žu w / ._f t t f t t R
t t
= − ⇒ = + > ∀ ∈
…utw†…uw†‡t†4~;:8
;4c•;4^(Q
7@8! ;\3‚!;5!•W3|!\%\?
8_]c8!\V…u;w(;;†.
 Nhận xét: fP
f
Dxxf
∈∀≥
_.wuŽ
8†…utw;;4
f
D
;c




=
=
⇔




=
=
.w‹u.w‹u
wuwu
yxF
yx
yxF
yfxf

H,N,8,|.*d2K-i42.
/ 0.
/ + > +x x− + − =
,8,
fP(Q%^
>
+
x
≥
_t‘;8!\V
/ 0.
00 10
/ 0.
/ / >
/ + > u w_ Žu w .
+
/ u /w ,U u+ >w
x x f x f x x
x x
− + − = = + > ∀ >

− −
_!88!\V;
>
+
x∀ ≥
\438!\V(~:;4
>
‰ ‹ w
+
+∞



mp;%_;4cT^!t†+fRt†+8^!a;W3
;4c
H,Q,8,|.*d2K-i42.
+ >
,u +w‰ u >w  u +wŠ /Uu /wu/wx x x x
− − + − = +
,8,
7%^t‡>_P(%;u/w
+ >
/Uu /w
u w  u >w  u +w u w
,u +w
x
f x x x g x
x
+
⇔ = − + − = =

−

#3T
+
/ /
Žu w ._ >
u >w + u +w >
U
Žu w ._ >
,u +w
f x x
x x
g x x
x
= + > ∀ >
− −
−
< ∀ >
−
fRPt‡>;c\V…utw(~:_8
utw`:mp;%…u//w†u//w†U_R;4cT^!a;
t†//
H,UMa;;4c
( )
+ > +
, + / / < /U /,x x x x x x
− − + > − + −
u<w
,8,
u<w

( ) ( )
+ >
+ / + / > + > <x x x x
 
⇔ − − + > − + −
 
( ) ( )
> >
+ / > + / + > +x x x x
⇔ − + − > − + −
”‘;8!\V…u;w†;
>
Œ>;_E†R
#3T…–u;w†>;
+
Œ+‡.…(~:;4R
( )
( )
+ / + + / +f x f x x x
− > − ⇔ − > −

”‘;t+•.;cd#^!({
”‘;t+
≥
.;c+t/‡.d#
+ / +x x
⇔ − > −
/x
⇔ > −
({

fR;R^!&†R
H,aMa;;4c
+
\
+
\+
>  + U .
<
>
x
x
 
 ÷
 
+ − ≥
u=w
,8,
#3T


+ +
+
\ \
\
+ + >
\+
 >  + U .  + U
< <
+
> >

\
>
+ +
+
\ \
/ \
+ > + /
  + U >  + U
< <
+ +
> >
\ +\
> >
x
x x
x
x
x
x x
x
x
   
 ÷  ÷
   
   
 ÷  ÷
   
+ − ≥ ⇔ + ≥
−
⇔ + ≥ ⇔ + ≥

7p;
[ ]
/‹._
+
\
∈=
txt
 da;;4c;4];8
+ U
<

0
/
>
>
+
≥+












tt

68!
tt
tf












+=
0
/
>
>
+
wu
`:P
[ ]
/‹.
∈∀
t

,w.uwu =≤⇒ ftf
m8

,+ U
<

>

&43_a;;4c(z"^!
nB}x}$$
H,B G8!\V
( )
+
+ >f x mx mx= + −
3#c!m(e;4cƒuxw=.T^!x∈‰/‹+Š
#c!m(ea;;4cƒuxw≤.^!({∀x∈‰/‹,Š
#c!m(ea;;4cƒuxw≥.T^!x∈
[ ]
/‹>
−
Giải:
SBd:(o;4cƒuxw=.;3T
( )
( )
( )
( )
+ +
+ +
> >
+ > . + >
+
/ /
f x mx mx m x x g x m

x x
x
= + − = ⇔ + = ⇔ = = =
+
+ −

7eƒuxw=.T^!x∈‰/‹+Š;c
[ ]
( )
[ ]
( )
/‹+
/‹+
m m3t
x
x
g x m g x
∈
∈
≤ ≤

>
/
1
m
⇔ ≤ ≤
nB#3T∀x∈‰/‹,Š;c
( )
+
+ > .f x mx mx= + − ≤

⇔
( )
+
+ >m x x+ ≤
⇔
( )
[ ]
+
>
_ /‹,
+
g x m x
x x
= ≥ ∀ ∈
+

[ ]
( )
/‹,
m 
x
g x m
∈
⇔ ≥

E
( )
( )
+
>

/ /
g x
x
=
+ −
!;4‰/‹,Š;⇔
[ ]
( )
( )
/‹,
/
m ,
1
x
g x g m
∈
= = ≥
0B#3TPx∈
[ ]
/‹>
−
;c
( )
+
+ > .f x mx mx= + − ≥
⇔
( )
+
+ >m x x+ ≥


7p;
( )
[ ]
+
>
_ /‹>
+
g x x
x x
= ∈ −
+
”‘;%A\3(@
Œ2:
.x
=
;ca;;4c;4];8
. . >m
= ≥
"^!
Œ2:
(
]
.‹>x∈
;cd#
⇔
( )
g x m
≤
T^!
(

]
.‹>x∈
(
]
( )
.‹>x
Min g x m
∈
⇔ ≤



E
( )
( )
+
>
/ /
g x
x
=
+ −
!-
(
]
.‹>
;
(
]
( ) ( )

.‹>
/
>
U
x
Min g x g m
∈
⇔ = = ≤
Œ2:
[
)
/‹.x∈ −
;c
+
+ .x x
+ <
d#
( )
g x m
⇔ ≥
T^!
[
)
/‹.x∈ −

[
)
( )
/‹.
Max g x m

−
⇔ ≥

#3T
( )
( )
( )
[ ]
+
+
> + +
._ /‹.
+
x
g x x
x x
− +
′
= ≤ ∀ ∈ −
+

E(T
( )
g x
`:;3T
[
)
( )
( )
/‹.

/ >Max g x g m
−
= − = − ≥
Kết luận:ƒuxw≥.T^!x∈
[ ]
/‹>
−
(
]
)
/
‹ > ‹
U
m

⇔ ∈ −∞ − +∞


U

H,B(Đề TSĐH khối A, 2007)
#c!m(e;4c
,
+
> / / + /x m x x
− + + = −
T^!;
Giải:
79
/x

≥
_:(o;4c
,
/ /
> +
/ /
x x
m
x x
− −
⇔ − + =
+ +

7p;
[
)
, ,
/
+
/ ._/
/ /
x
u
x x
−
= = − ∈
+ +

9(T
( )

+
> +g t t t m= − + =

#3T
( )
/
< + .
>
g t t t
′
= − + = ⇔ =

E(T‚
/
/
>
m
⇔ − < ≤
H,N. (Đề TSĐH khối B, 2007):G!4ƒfP!
.m
>
_;4c
( )
+
+ 1 +x x m x+ − = −
"T({3^!@^;
Giải:
7Q%^
+x
≥


d:(o;4c;3T
( ) ( ) ( )
+ < +x x m x⇔ − + = −
( ) ( ) ( )
+ +
+ < +x x m x⇔ − + = −
( )
( )
( )
> + > +
+ < >+ . +f t < >+x x x m x x x m
⇔ − + − − = ⇔ = = + − =

—;
( )
g x m
⇔ =
T({! ;^!; %
( )
+‹
+∞
#R;R;3T
( ) ( )
> , ._ +g x x x x
′
= + > ∀ >
E(T
( )
g x

(~:!8
( )
g x
;8
( )
( )
+ .‹ !
x
g g x
→+∞
= = +∞

( )
g x m
=
T({! ;^!∈
( )
+‹
+∞

fR
.m
∀ >
_;4c
( )
+
+ 1 +x x m x+ − = −
T3^!@^;
H,QB (Đề TSĐH khối D, 2007):
#c!m(e^;4cT^!

t01+0–0– 1


> >
> >
/ /
U
/ /
/U /.
x y
x y
x y m
x y

+ + + =



+ + + = −



Giải:
7p;
/ /
‹u x v y
x y
= + = +
;3T
(

)
(
)
>
>
>
/ / / /
> >x x x x u u
x x x
x
+ = + − × + = −
8
/ / / / /
+  +‹  +  +u x x x v y y
x x x y y
= + = + ≥ = = + ≥ =
9(T^;4];8
( )
> >
U
U
1
> /U /.
u v
u v
uv m
u v u v m
+ =

+ =



⇔
 
= −
+ − + = −



⇔
_u v
8^!W3;4cR3
( )
+
U 1f t t t m= − + =
6^T^!
( )
f t m
⇔ =
T+^!
/ +
_t t
;b3!z
/ +
+‹ +t t
≥ ≥

$Rd:;W38!\V
( )
f t

P
+t
≥
#
−∞
˜+ + U-+ Œ
∞
( )
f t
′
† † . ‡
( )
f t
Œ
∞
++
+
=-,
Œ
∞
2c:;;3T^T^!
=
+ ! ++
,
m
⇔ ≤ ≤ ∨ ≥
H,U#c!m(e
( ) ( )
> +
/

/ > ,
>
y x m x m x
−
= + − + + −
(~:;4u._>w
Giải.
68!\V;A;4u._>w⇔
( ) ( )
( )
+
+ / > . ._>y x m x m x
′
= − + − + + ≥ ∀ ∈
u/w
uEa†t43;! ;\V(e!n
( )
._>
∈
w
E
( )
y x
′
;;x=.8x=>u/w⇔y′≥.∀x∈‰._>Š
⇔
( )
[ ]
+
+ / + > ._>m x x x x+ ≥ + − ∀ ∈

⇔
( )
[ ]
+
+ >
._>
+ /
x x
g x m x
x
+ −
= ≤ ∀ ∈
+

[ ]
( )
._>
m3t
x
g x m
∈
⇔ ≤
#3T
( )
( )
[ ]
+
+
+ + 1
. ._>

+ /
x x
g x x
x
+ +
′
= > ∀ ∈
+
⇒guxw(~:;4‰._>Š⇒
[ ]
( ) ( )
._>
/+
m3t >
=
x
m g x g
∈
≥ = =


Nhận xét:Sử dụng phương pháp hàm số để giải toán là một trong những phương
pháp tối ưu khi giải các bài toán trong các đề thi đại học phần phương trình, hệ
phương trình và bất phương trình, đặc biệt là các bài toán tham số. Tuy nhiên, trong
phạm vi bài viết này tôi chỉ nêu một số ít bài toán để các em học sinh tham khảo. Tôi
hy vọng các em sẽ ứng dụng thành công những gì tôi đã truyền đạt trong đề tài để
đạt kết quả tốt trong quá trình học tập cũng như trong kỳ thi tốt nghiệp và các kỳ thi
tuyển sinh sắp tới.
B!E@
/”(`!(e;4c\3T^!


++,++
///+w+//u xxxxxm −−++−=+−−+
 u7_3(r%Vd˜+ ,w
+M;4c
xx
x
x
,
w/u
/+

+
+
+
−=
−
+
>M;4c
/+ =w/u
+ =
−=+
x
x
,#c!m (ea;;4c
mxxxx
+−≤−+
+w<wu,u
+
({

[ ]
<‹,
−∈∀
x
UMa;;4c
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Tân Phú, ngày 18 tháng 04 năm 2012
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