skkn_kythuatgiambientimgtnngtln
DESCRIPTION
giam bien tim GTLN-GTNNTRANSCRIPT
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K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 1
Bi ton tm gi tr nh nht (GTNN), gi tr ln nht (GTLN) ca mt biu
thc nhiu bin l mt bi ton bt ng thc v y l mt trong nhng dng ton kh
chng trnh ph thng. Trong thi tuyn sinh i hc, Cao ng hng nm, ni
dung ny thng xut hin dng cu kh nht. Trong Sch gio khoa Gii tch 12 th
ch trnh by cch tm GTNN, GTLN ca hm s (tc biu thc mt bin s). V vy,
mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha mt bin tr nn n
gin. Tuy nhin thc t, hu ht hc sinh l khng gii quyt c cho bi ton t hai
bin tr ln, thm ch cn c tm l khng c n.
Qua qu trnh ging dy lp chuyn Ton v luyn thi i hc ti tch ly
c mt s kinh nghim cho ni dung ny. Cc vn trnh by trong sng kin kinh
nghim l chuyn c ng dng trong ging dy lp 11T2 chuyn Ton ca
trng THPT chuyn Thoi Ngc Hu v cc lp luyn thi i hc. Sng kin kinh
nghim ny l s tng kt c chn ln cc chuyn ca bn thn vit ra trong thc
tin ging dy cng vi s ng gp nhit tnh ca qu Thy, C trong T Ton Tin
trng THPT chuyn Thoi Ngc Hu.
ti ny xut pht t nhng l do sau:
Gip hc sinh c thm kin thc v t tin hn trong vic gii quyt bi ton kh
ny.
Gip cho qu Thy, C v cc bn ng nghip dy Ton c mt ti liu tham
kho trong qu trnh ging dy b mn ca mnh. V qua chuyn ny ti hy
vng qu Thy, C v cc bn ng nghip s yu thch hn trong vic ging dy
chuyn ny. Thc t mt s Thy, C khng thch dy, v k c nhng Thy,
C nhiu nm luyn thi i hc cng khng i su lm v chuyn ny.
Phn m u
1. Bi cnh ca ti
2. L do chn ti
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 2
- ti ny c th p dng rng ri cho tt c gio vin dy Ton cc trng trung
hc ph thng tham kho v cc em hc sinh lp 12 n thi i hc, Cao ng.
- Phm vi nghin cu ca ti ny bao gm:
+ Nhc li cch tm GTNN, GTLN ca hm s thng qua mt vi v d.
+ H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha hai bin bng cch th mt bin qua bin cn li. + H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha hai bin bng cch t n ph theo tnh i xng t x y= + , 2 2t x y= + hoc t xy= . + H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha hai
bin bng cch t n ph theo tnh ng cp xty
= ..
+ H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha ba bin bng cch t n ph hoc th hai bin qua mt bin cn li.
Bn thn nghin cu ti ny nhm mc ch:
- Chia s vi qu Thy, C, cc bn ng nghip v cc em hc sinh kinh nghim
gii quyt bi ton tm GTNN, GTLN trong thi tuyn sinh i hc.
- Bn thn nhm rn luyn chuyn mn nhm nng cao nghip v s phm.
- Hng ng phong tro vit sng kin kinh nghim ca trng THPT chuyn Thoi
Ngc Hu.
Sng kin c chia thnh ba phn :
Phn m u Phn ni dung: gm 3 chng
Chng 1. Gi tr nh nht, gi tr ln nht ca hm s Chng 2. K thut gim bin trong bi ton tm GTNN, GTLN ca biu thc Chng 3. Mt s bi ton trong cc thi tuyn sinh i hc
Phn kt lun
3. Phm vi v i tng nghin cu
4. Mc ch nghin cu
5. Cu trc SKKN
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K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 3
Chng I
GI TR NH NHT , GI TR LN NHT CA HM S
Trong mc ny chng ti trnh by li mt s kin thc v o hm v mt s
cng thc v o hm.
1.1 nh l. Gi s D l mt khong hay hp cc khong.
Nu hai hm s ( )u u x= v ( )v v x= c o hm trn D th
( ) ;u v u v + = + ( ) ;u v u v =
( ) ;uv u v uv = + ( ) ;ku ku =
( ) 2u u v uvv v = , vi ( ) 0v x
1.2.nh l. o hm ca mt s hm s thng gp
( ) 0c = (c l hng s)
( ) 1x =
( ) ( )1n nx nx x = ( ) 1n nu nu u =
( ) 21 1x x = ( ) 21 uu u
=
( ) ( )12 0xx x = > ( ) 2u uu
=
( )x xe e = ( )u ue e u =
( ) ( )1ln 0xx x = > ( )ln uuu =
( )sin cosx x = ( )sin cosu u u =
( )cos sinx x = ( )cos sinu u u =
( ) ( )2 2tan 1 tanx x x k = + + ( ) ( )2tan 1 tanu u u = +
( ) ( )( )2t 1 cotco x x x k = + ( ) ( )2t 1 tco u u co u = +
Phn ni dung
I.1. Mt s kin thc c s v o hm
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 4
1.3 Nhn xt. o hm ca mt s hm phn thc hu t thng gp
1. Cho hm s ax bycx d
+=
+ vi . 0, 0a c ad cb . Ta c
( )2ad cb
cx dy
+ = .
2. Cho hm s 2ax bx c
ymx n+ +
=+
vi . 0a m . Ta c ( )
2
2
2b c
amx anxm n
mx ny
+ +
+ = .
3. Cho hm s 2
2
ax bx cy
mx nx p+ +
=+ +
vi . 0a m . Ta c ( )
2
22
2a b a c b c
x xm n m p n p
mx nx py
+ +
+ + = .
Trong mc ny chng ti trnh by li mt s kin thc v bi ton tm gi tr nh
nht, gi tr ln nht ca hm s.
2.1 nh ngha. Gi s hm s f xc nh trn tp hp D .
a) Nu tn ti mt im 0x D sao cho ( ) ( )0f x f x vi mi x D th s
( )0M f x= c gi l gi tr ln nht ca hm s f trn D , k hiu l
( )maxx D
M f x
= .
b) Nu tn ti mt im 0x D sao cho ( ) ( )0f x f x vi mi x D th s
( )0m f x= c gi l gi tr nh nht ca hm s f trn D , k hiu l
( )minx D
m f x
= .
2.1 Nhn xt. Nh vy, mun chng t rng s M (hoc m ) l gi tr ln nht (hoc
gi tr nh nht) ca hm s f trn tp hp D cn ch r :
a) ( )f x M (hoc ( )f x m ) vi mi x D ;
b) Tn ti t nht mt im 0x D sao cho ( )0f x M= (hoc ( )0f x m= ).
2.2 Nhn xt. Ngi ta chng minh c rng hm s lin tc trn mt on th t
c gi tr nh nht v gi tr ln nht trn on .
Trong nhiu trng hp, c th tm gi tr ln nht v gi tr nh nht ca hm
s trn mt on m khng cn lp bng bin thin ca n.
I.2. Gi tr nh nht, gi tr ln nht ca hm s
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 5
t 2 2 2
( )f t + 0 2 2 ( )f t 2 2
Quy tc tm gi tr nh nht, gi tr ln nht ca hm f trn on ;a b nh sau :
1. Tm cc im 1 2, ,..., nx x x thuc khong ( );a b m ti f c o hm bng 0
hoc khng c o hm.
2. Tnh ( ) ( ) ( ) ( )1 2, ,..., ,nf x f x f x f a v ( )f b .
3. So snh cc gi tr tm c.
S ln nht trong cc gi tr l gi tr ln nht ca f trn on ;a b , s nh nht
trong cc gi tr l gi tr nh nht ca f trn on ;a b .
Trong mc ny chng ti trnh by mt s v d tm gi tr nh nht, gi tr ln
nht ca hm s.
Th d 1 ( thi tuyn sinh i hc khi B 2003)
Tm gi tr nh nht v gi tr ln nht ca hm s ( ) 24f x x x= +
Li gii. Tp xc nh 2;2D = , ( ) 21 4x
f xx
=
, ( ) 0 2f x x = =
Bng bin thin
T bng bin thin ta c ( ) ( )2;2
min 2 2x
f x f
= = v ( ) ( )2;2
max 2 2 2x
f x f
= = .
Th d 2. ( thi tuyn sinh i hc khi B 2004)
Tm gi tr ln nht v nh nht ca hm s 2ln x
yx
= trn on 31;e
Li gii. Ta c ( )
2
2 2
12 ln . . ln ln 2 lnx x x x xxy
x x
= =
T c bng bin thin :
x 1 2e 3e y 0 + 0
y 0
24e
39e
I.3. Mt s th d tm GTNN, GTLN ca hm s
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K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 6
Vy ( ) 232 24
1;max
eey y e x e
= = = v ( )31;
min 1 0 1ey y x
= = =
Tm gi tr nh nht v gi tr ln nht ca hm s
1) ( ) ( )22 231 2 1f x x x= +
2) ( ) 5cos cos5f x x x= vi 4 4x
3) ( )4 2 2
2 2
2 1 1 1 3
1 1 1
x x xf x
x x
+ + + +=
+ + +
Hng dn. t 2 21 1t x x= + + , vi 2 2t .
Bi tp tng t
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 7
CHNG II
K THUT GIM BIN TRONG BI TON TM GI TR NH NHT, GI TR LN NHT CA BIU THC
T kt qu ca Chng I chng ta thy rng vic tm GTNN, GTLN ca hm
s kh n gin. Vic chuyn bi ton tm GTNN, GTLN ca mt biu thc khng t
hn hai bin sang bi ton tm GTNN, GTLN ca hm s cha mt bin s gip chng
ta gi c bi ton tm GTNN, GTLN ca mt biu thc.
Trong phn ny chng ti trnh by mt s dng bi ton tm GTNN, GTLN ca biu
thc cha hai bin bng cch th mt bin qua bin cn li. T xt hm s v tm
gi tr nh nht, gi tr ln nht ca hm s.
Th d 1. Cho , 0x y > tha mn 54
x y+ = . Tm gi tr nh nht ca biu thc
4 14
Px y
= +
Li gii. T gi thit 54
x y+ = ta c 54
y x= . Khi 4 15 4
Px x
= +
.
Xt hm s ( ) 4 15 4
f xx x
= +
vi 50;4
x
. Ta c ( )( )2 2
4 4
5 4f x
x x = +
.
Bng bin thin T bng bin thin ta c ( ) ( )
50;4
min 1 5x
f x f
= = .
Do min 5P = t c khi 11,4
x y= = .
x 0 1 54
( )f x 0 + + +
( )f x 5
II.1. Tm GTNN, GTLN ca biu thc bng phng php th
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 8
Th d 2. Cho ,x y tha mn 20, 12y x x y + = + . Tm gi tr nh nht, gi tr ln nht ca biu thc 2 17P xy x y= + + + .
Li gii. T gi thit 20, 12y x x y + = + ta c 2 12y x x= + v 2 12 0x x+ hay 4 3x . Khi 3 23 9 7P x x x= + . Xt hm s ( ) 3 23 9 7, 4;3f x x x x x = + . Ta c ( ) ( )2' 3 2 3f x x x= + .
Ta c bng bin thin
T bng bin thin ta c ( ) ( )4;3
min 1 12x
f x f
= = , ( ) ( ) ( )4;3
max 3 3 20x
f x f f
= = = .
Do min 12P = t c khi 1, 10x y= = v max 20P = t c khi
3, 6x y= = hoc 3, 0x u= = .
Th d 3. Cho , 0x y > tha mn 1x y+ = . Tm gi tr nh nht ca biu thc
1 1
x yP
x y= +
Li gii. T gi thit , 0x y > , 1x y+ = ta c 1 ,0 1y x x= < < .
Khi ta c 11
x xP
x x
= +
.
Xt hm s ( ) 11
x xf x
x x
= +
, ( )
( )2 1
2 1 1 2
x xf x
x x x x
+ =
.
Bng bin thin
T bng bin thin suy ra ( )
( )0;1
1min min 2
2xP f x f
= = = t c khi 1
2x y= = .
Nhn xt. Qua ba th d ny cho ta mt k thut gim bin khi tm GTNN, GTLN ca
biu thc hai bin bng cch th mt bin qua bin cn li v s dng cc gi thit
nh gi bin cn li. T tm GTNN, GTLN ca hm s cha mt bin b chn.
x 4 3 1 3 ( )f x + 0 0 +
20 20 ( )f x 13 12
x 0 12
1
( )f x 0 + + +
( )f x 2
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 9
1/ Cho , 3;2x y tha mn 3 3 2x y+ = . Tm gi tr nh nht, gi tr ln nht ca
biu thc 2 2P x y= + . 2/ Cho , 0x y tha mn 1x y+ = . Tm gi tr nh nht, gi tr ln nht ca biu
thc 1 1x y
Py x
= ++ +
3/ Cho , 0x y > tha mn 1x y+ = . Tm gi tr nh nht ca biu thc
2 22 2
1 1P x y
x y= + + +
4/ Cho 1x y+ = . Tm gi tr nh nht ca biu thc
( ) ( )3 3 2 23 3P x y x y x y= + + + + 5/ Cho , , ,a b x y tha mn 0 , 4a b< , 7a b+ v 2 3x y .
Tm gi tr nh nht ca biu thc ( )
2 2
2 2
2 2x y x yP
xy a b
+ + +=
+
Hng dn. Tm gi tr ln nht ca 2 2Q a b= + l M , xt hm s
( ) ( )2 22 2
,.
x y x yg y f x y
xy M+ + +
= = vi n y v x l tham s, tm gi tr nh nht
ca ( )g y l ( )h x . Sau tm gi tr nh nht ca hm s ( )h x vi 2;3x . 6/ Cho ,x y tha mn 3x y . Tm gi tr nh nht ca biu thc
2 2 8 16P x y x= + + . Hng dn. Nu 0x > th 6 2x y t xt hm s ( ) 6 2 8 16f x x x x= + + . Nu
0x th 2 2 8 16 16x y x+ + vi mi 30,x x y . 7/ Cho ( ), 0;1x y tha mn 1x y+ = . Tm gi tr nh nht ca biu thc
x yP x y= + .
Hng dn. Xt hm s ( ) ( ), 0;1xf x x x= . Chng minh ( ) ( )2 2
f x f y x yf + +
.
Ta c ( ) ( ) ( )1 11 1 2 22
xxP x x f x f x f = + = + =
.
8/ Cho , 0x y > tha mn 2x y+ = . Chng minh rng x yxy x y .
Bi tp tng t
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 10
Trong phn ny chng ti trnh by mt s dng bi ton tm GTNN, GTLN ca biu
thc cha hai bin m gi thit hoc biu thc th hin tnh i xng. T bng
php t n ph ta chuyn v bi ton tm G ca hm s.
Th d 1. Cho 2 2x y x y+ = + . Tm gi tr nh nht, gi tr ln nht ca biu thc 3 3 2 2P x y x y xy= + + + Li gii.
t t x y= + , t gi thit 2 2x y x y+ = + ta c ( ) ( )2 22xy x y x y t t= + + =
hay 2
2t t
xy
= . p dng bt ng thc ( ) ( ) ( )2 2 22 2x y x y x y+ + = + hay 2 2t t suy ra 0 2t . Khi biu thc ( ) ( )3 22P x y xy x y t= + + = . Do ta
c max 4P = t c khi 2t = hay 2x y+ = v 1xy = suy ra 1x = v 1y = , ta c min 0P = t c khi 0t = hay 0x y= = .
Nhn xt. Bi ton ny gi thit v biu thc P c cho di dng i xng vi hai
bin. V vy, chng ta ngh n cch i bin t x y= + . Nhng gii bi ton trn
vn th phi tm iu kin ca bin t . Sau y l mt s bi ton vi nh hng tng
t.
Th d 2. Cho , 0x y > tha mn 2 2 1x xy y+ + = .
Tm gi tr ln nht ca biu thc 1
xyP
x y=
+ +
Li gii. t t x y= + . T gi thit , 0x y > v 2 2 1x xy y+ + = suy ra 2 1xy t= .
p dng bt ng thc ( )2 4x y xy+ suy ra 103
t< .
Khi 3 313
P t
= .
V vy 3 3max3
P
= t c khi ( ) 2 1; ;3 3
x y =
hoc ( ) 1 2; ;3 3
x y =
.
Th d 3. Cho ,x y tha mn 1x y+ v 2 2 1x y xy x y+ + = + + . Tm gi
tr nh nht, gi tr ln nht ca biu thc 1
xyP
x y=
+ +.
II.2. Tm GTNN, GTLN ca biu thc c tnh cht i xng
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 11
Li gii. t t x y= + . T gi thit 2 2 1x y xy x y+ + = + + ta c
( ) ( )2 1x y xy x y+ = + + hay 2 1xy t t= .
p dng bt ng thc ( )2 4x y xy+ suy ra 23 4 4 0t t hay 2 23t . Khi
2 1
1t t
Pt
=+
. Xt hm s ( )2 1
1t t
f tt
=+
, ( )( )
2
2
2
1
t tf t
t
+ =+
,
( ) 0f t = 0 2t t = = (loi). Bng bin thin T bng bin thin ta c ( ) ( )
23;2
min min 0 1t
P f t f
= = = t c khi
( ) ( ); 1;1x y = hoc ( ) ( ); 1; 1x y = v ( ) ( ) ( )23
23;2
1max max 2
3tP f t f f
= = = =
t c khi 13
x y= = hoc 1x y= = .
Th d 4. Cho ,x y tha mn 0 , 1x y< v 4x y xy+ = . Tm gi tr nh nht,
gi tr ln nht ca biu thc 2 2P x y xy= + .
t t x y= + . T gi thit 0 , 1x y< v 4x y xy+ = suy ra 4t
xy = v 1 2t .
Khi ( )2 2 334
P x y xy t t= + = . Xt hm s ( ) 2 34
f t t t= , ( ) 324
f t t = ,
( ) 0f t = 38
t = (loi). Bng bin thin
T bng bin thin ta c ( ) ( )1;2
1min min 1
4tP f t f
= = = t c khi 1
2x y= = v
( ) ( )1;2
5max max 2
2tP f t f
= = = t c khi ( ) ( )2 22 22 2; ;x y += hoc
( ) ( )2 2 2 22 2; ;x y + = .
t 23 0 2 ( )f t 0 +
( )f t 13
1
13
t 1 2 ( )f t +
( )f t 14
52
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 12
Th d 5. Cho ,x y tha mn , 0x y v ( ) 2 2 2xy x y x y x y+ = + + . Tm
gi tr ln nht ca biu thc 1 1Px y
= + .
Li gii. t t x y= + . T gi thit ( ) 2 2 2xy x y x y x y+ = + + hay
( ) ( ) ( )2 2 2xy x y x y xy x y+ = + + + suy ra 2 2
2t t
xyt +
=+
. p dng bt ng
thc ( )2 4x y xy+ suy ra 3 22 4 8
02
t t tt
+
+ hay 2 2t t< . Khi
2
2
22
x y t tP
xy t t
+ += =
+. Xt hm s ( )
2
2
22
t tf t
t t
+=
+, ( )
( )
2
22
3 4 4
2
t tf t
t t
+ + = +
,
( ) 0f t = 232t t = = (loi). Bng bin thin T bng bin thin ta c ( ) ( )
2 2max max 2 2
t tP f t f
tha mn 3xy x y+ + = .
Chng minh rng 2 23 3 31 1 2x y xy
x yy x x y
+ + + ++ + +
Li gii. t t x y= + . T gi thit , 0x y > , 3xy x y+ + = v p dng bt ng
thc ( )2 4x y xy+ ta c 3 , 0xy t t= > v 2 4 12 0t t+ hay 2t hoc 6t (loi). Khi bt ng thc cn chng minh tr thnh
( ) ( )( ) ( )
223 6 3 32
1 2
x y xy x y xyx y xy
xy x y x y
+ + ++ + +
+ + + + hay
22 3 23 9 18 3 92 4 12
4 2t t t
t t t t tt
+ + + + ( )( )22 6 0t t t + +
lun ng vi mi 2t , du bng xy ra khi 2t = hay 1x y= = .
Th d 7. Cho , 0x y > tha mn 2 2 1x y+ = . Tm gi tr nh nht ca biu thc
( ) ( )1 11 1 1 1P x yy x
= + + + + + .
Li gii. t t x y= + . T gi thit , 0x y > v 2 2 1x y+ = suy ra 2 12t
xy
= v
1t > . p dng bt ng thc ( ) ( )2 2 22x y x y+ + suy ra 1 2t< .
t 2 23 2 +
( )f t 0 + 0 _
( )f t 1
0
2
1
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 13
Khi ( )2
11
x y t tP x y xy
xy t
+ + = + + + = . Xt hm s ( )
2
1t t
f tt+
=
,
( )( )
2
2
2 1
1
t tf t
t
=
, ( ) 0 1 2f t t = = (loi). Bng bin thin
T bng bin thin ta c
(( ) ( )
1; 2min min 2 4 3 2
tP f t f
= = = + t c khi
1
2x y= = .
Nhn xt. Qua cc th d trn, cho ta mt k thut gim bin ca bi ton tm GTNN,
GTLN ca biu thc hai bin c tnh i xng: Do tnh i xng nn ta lun c th
bin i a v mt trong cc dng t t x y= + , 2 2t x y= + hoc t xy= , t a
v tm GTNN, GTLN ca hm s.
1/ Cho , 0x y > tha mn 1 3x y xy+ + = . Tm gi tr ln nht ca biu thc
( ) ( ) 2 23 3 1 11 1
x yP
y x x y x y= +
+ +
2/ Cho ,x y khng ng thi bng 0 v tha mn 1x y+ = . Tm gi tr nh nht ca
biu thc 2 2
2 2 2 2
11 1
x yP
x y y x= + +
+ + +
3/ Cho 2 2 1x y+ = . Tm gi tr nh nht, gi tr ln nht ca biu thc
1 1P x y y x= + + + 4/ Cho 2 2 1x y+ = . Tm gi tr nh nht, gi tr ln nht ca biu thc
1 1
x yP
y x= +
+ +
5/ Cho , 0x y thay i tha mn ( ) 2 2x y xy x y xy+ = + . Tm gi tr ln nht ca
biu thc 3 3
1 1P
x y= + .
6/ Cho ,x y tha mn 2 2 2x xy y+ + . Tm gi tr ln nht ca biu thc 2 2P x xy y= + .
t 1 2
( )f t
( )f t
+
4 3 2+
Bi tp tng t
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 14
Trong phn ny chng ti trnh by mt s dng bi ton tm gi tr nh nht, gi tr ln nht ca biu thc cha hai bin m gi thit hoc biu thc th hin tnh ng cp. T xt hm s v tm gi tr nh nht, gi tr ln nht ca hm s.
Th d 1. Cho , 0x y > tha mn 2 2 1x y+ = . Tm gi tr ln nht ca biu thc
( )P y x y= + .
Li gii. t y tx= . T iu kin , 0x y > suy ra 0t > . T gi thit 2 2 1x y+ = ta
c 22
11
xt
=+
. Khi biu thc ( )2
22
11
t tP x t t
t
+= + =
+. Xt hm s ( )
2
2 1t t
f tt
+=
+,
( )( )
2
22
2 1
1
t tf t
t
+ + =+
, ( ) 0 2 1 1 2f t t t = = + = (loi). Bng bin thin
T bng bin thin ta c ( ) ( ) 2 120max max 2 1tP f t f
+
>= = + = t c khi
( ) ( )2 12 12 2 2 2; ;x y += .
Th d 2. Cho , 0x y v tha mn 2 2 1x y+ = . Tm gi tr ln nht ca biu thc 2
2
4 6 52 2 1x xy
Pxy y
+ =
Li gii.
Nu 0x = th t gi thit 2 2 1x y+ = v 0y suy ra 1y = . Khi 53
P = .
Nu 0x th t y tx= . T gi thit , 0x y v 2 2 1x y+ = suy ra 0t v
22
11
xt
=+
. Khi ( )( )2 2
2 2 2
4 6 5 5 6 12 2 1 3 2 1
x t t tP
x t t t t
+ += =
+. Xt hm s
( )2
2
5 6 13 2 1t t
f tt t
+=
+, ( )
( )
2
22
8 4 4
3 2 1
t tf t
t t
+ = +
, ( ) 10 12
f t t t = = = (loi).
Bng bin thin
t 0 2 1+ +
( )f t + 0
( )f t
0
2 12+
1
II.3. Tm GTNN, GTLN ca biu thc c tnh ng cp
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 15
T bng bin thin ta c ( ) 5 , 03
f t t< v ( ) ( )120min min 1tP f t f= = = t
c khi 25
x = v 15
y = . V vy 5max3
P = t c khi ( ) ( ); 0;1x y = v
min 1P = t c khi ( ) ( )2 15 5; ;x y = .
Th d 2. Cho , 0x y > . Chng minh rng ( )
2
32 2
4 184
xy
x x y
+ +
Li gii. t xty
= . T gi thit , 0x y > suy ra 0t > . Khi bt ng thc cn
chng minh tng ng vi ( )32
4 184
t
t t
+ + hay ( )32 4 2t t t+ . Xt hm
s ( ) ( )32 4f t t t t= + ,
( ) ( ) ( ) ( ) ( )3 3
2 2 23
2
2 2
3 4 4 4 34
4 4
t t t t t t tf t t t
t t
+ + + = + =
+ +,
( ) 220f t t = = . Ta c bng bin thin
T bng bin thin ta c ( ) ( )220max 2t f t f> = = hay ( )3
2 4 2t t t+ du bng
xy ra khi 22
t = hay 2y x= .
t 0 22
+ ( )f t + 0
( )f t 0
2
0
t 0 12 +
( )f t 0 +
( )f t
1 1
53
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 16
1/ Cho , 0x y > tha mn 1xy y . Tm gi tr nh nht ca biu thc
2 3
2 39
x yP
y x= +
2/ Cho , 0x y . Chng minh rng 3 3 23 7 9x y xy+ .
3/ Cho , 0x y . Chng minh rng 4 4 3 3x y x y xy+ + .
4/ Tm gi tr nh nht ca biu thc 2 2
2 23 8x y x y
Py xy x
= + + vi , 0x y .
Trong phn ny chng ti trnh by mt s dng bi ton tm gi tr nh nht, gi tr
ln nht ca biu thc cha ba bin bng cch t n ph hoc th hai bin qua mt
bin cn li. T , chuyn c bi ton v bi ton tm gi tr nh nht, gi tr ln
nht ca hm s.
Th d 1. Cho , , 0x y z > tha mn 1x y z+ + = . Chng minh rng 1 1 16xz yz
+
Li gii. t t x y= + . T gi thit ta c ( )1 1z x y t= + = v 0 1t< < .
p dng bt ng thc ( )2 4x y xy+ hay 2
4t
xy .
Khi ( ) 2
1 1 41t
Pxz yz xy t t t
= + = +
.
Xt hm s ( ) 24
f tt t
= +
, ( ) ( )( )224 2 1t
f tt t
=
+, ( ) 10
2f t t = = .
Ta c bng bin thin
t 0 12 1
( )f t 0 +
( )f t
+
16
+
Bi tp tng t
II.4. Tm GTNN, GTLN ca biu thc ba bin
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 17
T bng bin thin ta c ( )
( ) ( )120;1min 16t f t f = = t c khi 1 14 2,x y z= = = .
V vy 1 1 16xz yz
+ .
Th d 2. Cho 2 2 2 1x y z+ + = . Tm gi tr nh nht, gi tr ln nht ca biu thc P x y z xy yz zx= + + + + +
Li gii. t t x y z= + + . p dng bt ng thc Cauchy Schwarz ta c
( ) ( )2 2 2 23x y z x y z+ + + + suy ra 3 3t . Khi
( ) ( ) ( ) ( )2 2 2 2 21 1 2 12 2
P x y z x y z x y z t t = + + + + + + + = +
Xt hm s ( ) ( )21 2 12
f t t t= + , ( ) 2 2f t t = + , ( ) 0 1f t t = = . Ta c bng bin thin T bng bin thin ta c ( ) ( )
3; 3min min 1 1
tP f t f
= = = t c khi 1t =
hay ( ) ( ); ; 1;0;0x y z = v cc hon v ca n; ( ) ( )
3; 3max max 3 1 3
tP f t f
= = = + t c khi 3t = hay
( ) ( )1 1 13 3 3; ; ; ;x y z = .
Th d 3. Cho , , 0x y z tha mn 1x y z+ + = .
Chng minh rng 3 3 3 15 14 4
x y z xyz+ + +
Li gii. Do vai tr ca , ,x y z bnh ng nn ta lun gi s c { }min , ,x x y z= . T
gi thit , , 0x y z , 1x y z+ + = ta c 103
x v 1y z x+ = . p dng bt
ng thc ( )2
4
y zyz
+ v 27 3 0
4x < . Khi biu thc
( ) ( )33 3 3 315 1534 4
P x y z xyz x y z yz y z xyz= + + + = + + + +
( ) ( ) ( )3 33 315 273 1 34 4x x
x y z yz y z x x yz = + + + + = + +
t 3 1 3
( )f t 0 +
( )f t 1 3
1
1 3+
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 18
( ) ( ) ( )2
33 3 227 11 3 27 18 3 44 4 16
y z xx x x x x
+ + + = + +
Xt hm s ( ) ( )3 21 27 18 3 416
f x x x x= + + , ( ) ( )21 81 36 316
f x x x = + ,
( ) 109
f x x = =13
x = . Bng bin thin
T bng bin thin ta c ( ) ( ) ( )13
130;
1max 0
4xf x f f
= = = . Do 14
P . Du bng xy
ra khi ( ) ( )1 1 13 3 3; ; ; ;x y z = hoc ( ) ( )1 12 2; ; 0; ;x y z = v cc hon v ca n.
Th d 4. Cho ( ), , 0;1x y z tha mn 1xy yz zx+ + = .
Tm gi tr nh nht ca biu thc 2 2 21 1 1
x y zP
x y z= + +
.
Li gii. Ta c ( ) ( ) ( )
2 2 2
2 2 21 1 1x y z
Px x y y z z
= + +
. Xt hm s ( ) ( )211
f tt t
=
vi 0 1t< < , ( )( )
2
22 2
3 1
1
tf t
t t
=
, ( ) 13
10
3f t t t = = = (loi).
Ta c Bng bin thin
T bng bin thin ta c ( ) ( )21 3 3
0;121t
t t
.
V vy ( ) ( )2 2 23 3 3 3 3 32 2 2
P x y z xy yz zx + + + + =
Do 3 3min2
P = t c khi 13
x y z= = = .
x 0 19
13
( )f x + 0 0
( )f x 14
727
14
t 0 13
1
( )f t 0 +
( )f t +
3 32
+
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 19
Chng III
MT S BI TON TRONG CC THI I HC
Bi 1 ( thi tuyn sinh i hc A 2011)
Cho , ,x y z l ba s thc thuc on 1;4 v ,x y x z . Tm gi tr nh nht ca
biu thc 2 3x y z
Px y y z z x
= + ++ + +
Li gii. Trc ht ta chng minh : 1 1 21 1 1a b ab
+ + + +
(*), vi a v b dng
v 1ab . Tht vy, ( ) ( )( ) ( )( )* 2 1 2 1 1a b ab a b + + + + +
( ) 2 2a b ab ab a b ab + + + + ( )( )21 0ab a b , lun ng vi ,a b dng v 1ab . Du bng xy ra, khi v ch khi : a b= hoc 1ab = .
p dng (*), vi x v y thuc on 1;4 v x y , ta c:
1 1 1 22 3 3
1 1 2 1
xP
x y z x y xy z x y
= + + ++
+ + + +
Du bng xy ra, khi v ch khi : z xy z= hoc 1x
y= (1)
t , 1;2x t ty
= . Khi : 2
2
212 3
tP
tt +
++. Xt hm s :
( )2
2
2, 1;2
12 3t
f t ttt
= + ++, ( )
( ) ( )
( ) ( )
3
2 22
2 4 3 3 2 1 90
2 3 1
t t t tf t
t t
+ + = , suy ra : )( )
52;
5 23min
2 4f t f
+
= = .
Vy, 23min4
P = t khi v ch khi : 52
a bb a+ =
v 1 12a ba b
+ = + ( ) ( ); 2;1a b = hoc ( ) ( ); 1;2a b =
Bi 3 ( thi tuyn sinh i hc khi B-2010)
Cho cc s thc khng m , ,a b c thon mn 1a b c+ + = . Tm gi tr nh nht ca
biu thc ( ) ( )2 2 2 2 2 2 2 2 23 3 2M a b b c c a ab bc ca a b c= + + + + + + + +
Li gii. Ta c: ( ) ( ) ( )2 3 2 1 2M ab bc ca ab bc ca ab bc ca + + + + + + + +
t t ab bc ca= + + , ta c : ( )2 1
03 3
a b ct
+ + = .
Xt hm s ( ) 2 3 2 1 2f t t t t= + + trn 10;2
, ta c : ( ) 22 3
1 2f t t
t = +
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 21
( )( )32
2 01 2
f tt
=
, du bng ch xy ra ti 0t = , suy ra ( )f t nghch bin.
Xt trn on 10;3
ta c : ( ) 1 11 2 3 03 3
f t f = >
, suy ra ( )f t ng bin. Do
: ( ) ( ) 10 2, 0;3
f t f t =
.
V th : ( ) 12, 0;3
M f t t
. 2 , 1M ab bc ca ab bc ca= = = + + = v
1a b c+ + = ( ); ;a b c l mt trong cc b s : ( ) ( ) ( )1;0;0 , 0;1;0 , 0;0;1
Do gi tr nh nht ca M l 2.
Bi 4. ( thi tuyn sinh i hc khi B 2009)
Tm gi tr nh nht ca biu thc ( ) ( )4 4 2 2 2 23 2 1A x y x y x y= + + + + vi ,x y l
cc s tha mn ( )3 4 2x y xy+ + .
Li gii. Da vo bt ng thc hin nhin : ( )2 4x y xy+ nn
( ) ( ) ( ) ( )3 3 2 34 2 4 2x y xy x y x y x y xy+ + + + + + +
( ) ( )3 2 2 0x y x y + + + ( ) ( ) ( )21 2 0x y x y x y + + + + + (1)
Do ( ) ( ) ( )2
2 1 72 0
2 4x y x y x y
+ + + + = + + + >
v t (1) suy ra : 1x y+ .
Vy nu cp ( );x y tha mn yu cu bi th 1x y+ (2).
Ta bin i A nh sau:
( ) ( )4 4 2 2 2 23 2 1A x y x y x y= + + + +
( ) ( ) ( )22 2 4 4 2 23 3 2 12 2x y x y x y= + + + + + (3)
Do ( )22 24 4
2
x yx y
++ nn t (3) suy ra :
( ) ( ) ( ) ( ) ( )2 2 22 2 2 2 2 2 2 2 2 23 3 92 1 2 12 4 4
A x y x y x y x y x y + + + + + = + + +
V ( )22 22
x yx y
++ nn t (2) ta c : 2 2 1
2x y+ .
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 22
t ( ) 29 2 14
f t t t= + vi 2 2 12
t x y= + . Ta c : ( ) 9 12 0,2 2
f t t t = > .
Suy ra : ( )12
1 9min
2 16tf t f
= = (4)
T (4) suy ra 916
A . Mt khc d thy khi 12
x y= = th 916
A = .
Vy 9min16
A = khi 12
x y= = .
Bi 5. ( thi tuyn sinh i hc khi D 2009)
Cho , 0x y v 1x y+ = . Tm gi tr ln nht v nh nht ca biu thc :
( )( )2 24 3 4 3 25S x y y x xy= + + +
Li gii. Ta c : ( )( ) ( )2 2 2 2 3 34 3 4 3 25 16 12 34S x y y x xy x y x y xy= + + + = + + +
( )( )2 2 2 216 12 34x y x y x xy y xy= + + + + ( )22 216 12 3 34x y x y xy xy = + + + 2 216 2 12x y xy= + (1) (do 1x y+ = ).
t xy t= . Vi 0, 0x y , ta c : ( )2 1 1
0 04 4 4
x yxy t
+ =
Xt hm s ( ) 216 2 12f t t t= + vi 104
t . Ta c : ( ) 32 2f t t = . Bng bin thin
Suy ra ( ) ( )14
1160;
191min
16f t f
= = v ( ) ( )14
140;
25max
2f t f
= = .
Vy: Gi tr nh nht ca S t c
2 3 2 31 ;1 4 4116 2 3 2 3;16 4 4
x y x yt
xyx y
+ + = = = = += = =
Gi tr ln nht ca S t c 1
1 114 24
x yt x y
xy
+ = = = = =
t 0 116
14
( )f t 0 +
( )f t 12
19116
252
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 23
Sng kin ny t c mt s kt qu sau :
+ Nhc li cch tm GTNN, GTLN ca hm s thng qua mt vi v d.
+ H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha hai bin
bng cch th mt bin qua bin cn li.
+ H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha hai bin
bng cch t n ph theo tnh i xng t x y= + , 2 2t x y= + hoc t xy= .
+ H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha hai bin
bng cch t n ph theo tnh ng cp xty
= ..
+ H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha ba bin
bng cch t n ph hoc th hai bin qua mt bin cn li.
Qua thc t ging dy chng ti thy rng vn no d kh m gio vin quan tm
v truyn th cho hc sinh bng lng say m v nhit tnh ca mnh th s cun ht cc
em vo con ng nghin cu. Bi ton tm GTNN, GTLN ca mt biu thc khng
phi l mt vn mi, nhng thc t cho thy cn nhiu Thy, C cha quan tm
ng mc vn ny.
Vi sng kin kinh nghim ny hy vng gp thm mt ti liu cho qu Thy, C v
cc bn ng nghip ; gip cc em hc sinh c thm nhng kinh nghim cho loi ton
ny, t t tin hn khi thi i hc.
Phn kt lun
2. Bi hc kinh nghim
3. ngha ca SKKN
1. Kt qu t c
-
K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc
Gio vin: Trn Phi Thon 24
Sng kin kinh nghim ny c th trin khai nh mt chuyn bi dng hc sinh
gii ; cng nh dng ging dy cho cc em hc sinh n tp thi i hc, nhm gip
cc em hc sinh c th vt qua tr ngi tm l t trc ti nay cho loi bi ton ny.
1. Sch Gio khoa Gii tch 12, Nh xut bn Gio dc Vit Nam.
2. Tuyn tp Tp ch Ton hc v tui tr nm 2008, 2009, 2010 ; v Tp ch Ton hc
v tui tr hng thng.
3. Ngun Internet : http://www.VnMath.com.vn,
4. Kh nng ng dng v trin khai
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