khai thac tinh doi xung trong chung minh bdt
DESCRIPTION
BDTTRANSCRIPT
Cu 1 (3 im) Gii cc phng trnh sau:
Phn 2
GII QUYT VN I. C s l thuyt
1) a thc c gi l i xng i vi x v y nu
* Tnh cht.
+ Mi a thc i xng u biu din c qua v
+
2) Nu hm s c v
th
Chng minh.
TH 1. a = 0. Kt qu l hin nhin
TH 2. a > 0 th l hm s ng bin trn on nn
TH 3. a < 0 th l hm s nghch bin trn on nn
Vy
3. Php chun ha.Biu thc c gi l thun nht bc n nu
Do vy, nu k > 0 th . t
v chn k = > 0 th v .
Do i vi nhng bt ng thc thun nht ta c th gi thit thm (hoc nu chn ). Vic lm trn gi l chun ha.
II. Bi tp p dng
Bi 1 (IMO 1984)
Cho x, y, z l cc s thc khng m tha mn . Chng minh rng
(1)
Gii.
(1)
.
t v xt
Ta c
. ng thc xy ra khi v ch khi . Bt ng thc c chng minh.
Nhn xt. Trong bi ton ny, ta khng bit c du ca . Tuy nhin, ta c th kt lun c gi tr nh nht ca trn on ch t c ti hai u mt. Cch lm ny trnh c vic phi xt nhiu trng hp.Bi 2 (Olympic 30 4 nm 2000)
Cho a, b, c l cc s thc khng m tha mn . Chng minh rng
(2)Gii.
(2)
t v xt
Ta c
. ng thc xy ra khi v ch khi hoc v cc hon v ca n. Bt ng thc c chng minh.Nhn xt. Trong cc chng minh trn, gi thit l quan trng. V vy i vi nhng bt ng thc cha cho gi thit ny m c tnh ng bc (thun nht) th ta c th chun ha to ra chng.
Bi 3 (BT AM GM cho ba s khng m)
Cho a, b, c l cc s thc khng m. Chng minh rng
Chng minh.
TH 1. . BT l hin nhin.
TH 2. . Chia hai v cho ta c
t th v bt ng thc cho tr thnh hay .t v xt hm s trn
Ta c ; (do ).Vy . BT c chng minh.
ng thc xy ra khi v ch khi hay
Nhn xt. BT trn l thun nht nn ta c th to ra gi thit . T y, nu BT l thun nht th ta c th gi thit thm .Bi 4 (IMO 1964)
Cho x, y, z l cc s thc khng m. Chng minh rng
(3)Gii.
Do BT l thun nht nn nh chun ha ta c th gi thit thm .Thay vo (3) ta c
t v xt
Ta c
.
ng thc xy ra hoc v cc hon v ca n. iu ny tng ng vi hoc v cc hon v ca n.Bt ng thc c chng minh.
Bi 5. Cho x, y, z l cc s thc khng m. Chng minh rng
(4)
Gii.Do BT l thun nht nn nh chun ha ta c th gi thit thm .Khi (4)
EMBED Equation.DSMT4
. Theo Bi 4 th BT ny l ng.Bi 6. Cho a, b, c tha mn . Chng minh rng .Gii. Coi c nh tham s, ta c h i xng loi (I) i vi a, b
Chng minh tng t ta c . Vy
Bi 7. Trong cc nghim (x ; y) ca h , hy tm nghim sao cho t gi tr ln nht, gi tr nh nht.
Gii. t th h tng ng vi
Ta c
iu kin i vi a h c nghim (x ; y) l
(tha mn iu kin ).Khi
Xt hm s .
Lp bng bin thin ca hm s ny trn on ta c
khi hoc , tc l khi hoc
khi , tc l khi hoc
Bi 8 (HSG Quc gia nm 2005)
Cho x, y tha mn . Tm gi tr ln nht v gi tr nh nht ca x + y.
Gii. t . Bi ton tr thnh: tm S h phng trnh sau c nghim
(I)
t th v . H tr thnh
H (I) c nghim (x ; y) khi v ch khi h (II) c nghim (a ; b) sao cho a ( 0, b ( 0
Vy
Bi 9 ( thi tuyn sinh i hc nm 2006 khi A)
Cho hai s thc thay i v tha mn: .
Tm gi tr ln nht ca biu thc A = .
Gii.
Ta c . t .
Bi ton tr thnh: Cho a, b thay i v tha mn . Tm gi tr ln nht ca A = . Ta c A = .T gi thit .
t v xt h phng trnh
.
ng thc xy ra . Vy .III. Bi tp tham kho
1. Cho a, b, c l cc s thc khng m tha mn . Chng minh rng
2. Cho x, y, z l cc s thc dng tha mn . Chng minh rng
3. Cho x, y, z l cc s thc dng tha mn . Chng minh rng
4. Cho x, y, z l cc s thc dng tha mn . Chng minh rng
5. Cho hai s thc x, y thay i tha mn . Tm gi tr ln nht v gi tr nh nht ca biu thc S =
6. Cho cc s thc dng x, y tha mn . Tm gi tr ln nht ca biu thc
P =
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