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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
1
L XUN I
(GV THPT Chuyn Vnh Phc)
Bt ng thc (BT) l mt trong nhng dng ton thng c trong cc thi H-
C. Cc th sinh ca chng ta u rt s v lng tng khi gp phi bi ton chng minh
BT hoc tm gi tr ln nht, nh nht. n gin l do cc bi ton v BT thng l bi
ton kh trong thi, nhm phn loi v chn c cc hc sinh kh gii. Thng th cc s
t khng bit bt u t u gii quyt cc bi ton v BT. Chuyn ny mun h
thng cho cc bn cc phng php c bn v mt s dng bi tp v BT. Hy vng s
gip cc em hc sinh lp 12 t kt qu cao trong k thi H- C sp ti.
c xong chuyn ny ti tin cc bn s khng cn cm gic s bt ng thc na
Khi chng ta ht i s s hi v ngi ngn th chng ta s am m v dnh tnh yu cho n.
Dnh tnh yu v s am m cho ton hc ni chung v BT ni ring l iu rt cn thit
ca mt ngi lm ton s cp chn chnh v s lng mn ca ton hc cng bt ngun t
Thnh cng ch n khi bn lm vic tn tm v lun ngh n nhng iu tt
p
Nhng li khuyn b ch khi hc v BT:
1. Nm chc cc tnh cht c bn ca BT.
2. Nm vng cc phng php c bn chng minh BT nh: PP bin i tng
ng; PP s dng BT C si; PP s dng o hm
3. c bit ch trng vo n tp cc k thut s dng BT C si, lun bit t v tr
li cc cu hi nh: khi no p dng; iu kin cho cc bin l g; du bng xy ra khi no;
nu p dng th th c xy ra du bng khng; ti sao li thm bt nh vy
4. Lun bt u vi cc BT c bn (iu ny v cng quan trng); hc thuc mt
s BT c bn c nhiu p dng nhng phi ch iu kin p dng c, chng hn nh:
* 2 2 2a b c ab bc ca (1) vi mi a,b,c
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
2
* 2(a b c) 3(ab bc ca) (2) vi mi a,b,c
* 2 2 2 2(a b c) 3(a b c ) (3) vi mi a,b,c
* 1 1 4 1 1 1 9; (4)a b a b a b c a b c
vi mi a,b,c dng
* 2 2 2 2 2 2a x b y (a b) (x y) (5) vi mi a,b,x,y.
* 2 2 2x y (x y) (6)
a b a b
vi mi a,b dng v x,y bt k
* 2 2 2 2x y z (x y z) (7)
a b c a b c
vi mi a,b,c dng v x,y,z bt k
Du bng xy ra cc BT (1), (2), (3) v (4) l a=b=c.
Du bng xy ra BT (5) v (6) l x ya b ; (7) l x y z
a b c (vi mu khc 0).
(Cc em hy bt tay ngay vo vic chng minh cc BT c bn trn nh. Hy tm cho mnh
mt cch gii nht qun, n gin, nh n v khi lm bi thi u phi chng minh li, ri
mi c p dng).
Trc ht xin a ra 3 phng php thng dng nht chng minh BT
I. PHNG PHP BIN I TNG NG:
1. Phng php chung
chng minh A B ta thng thc hin theo mt trong hai cch sau:
Cch 1: Ta chng minh A B 0 . lm c iu ny ta thng s dng hng ng
thc phn tch A B thnh tng hoc tch ca nhng biu thc khng m.
Cch 2: Xut pht t mt BT ng no ta bin i n BT cn chng minh. i vi
cch ny thng cho ta li gii khng c t nhin cho lm v thng s dng khi cc
bin c nhng rng buc c bit.
Ch : Mt s kt qu hay s dng
* 2x 0 vi mi x v 2x 0 x 0
* x 0 vi mi x v x 0 x 0
2. Mt s v d
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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V d 1: Chng minh rng vi mi a,b ta c: 2 2a b 2ab (1)
Gii: Ta c 2 2 2 2 2a b 2ab (a b) 0 a b 2ab (pcm).
ng thc xy ra khi v ch khi a=b.
Tht n gin phi khng cc bn, nu tinh thm mt cht thi cc bn s tm ra nhng
kt qu tng qut hn v nim tin vt qua bi BT trong thi H l hon ton kh
thi.
C th l vi ba s thc a,b,c bt k ta c 2 2a b 2ab ; 2 2b c 2bc v 2 2a c 2ac
Cng tng v ca 3 BT ta c kt qu sau: 2 2 2a b c ab bc ca (2)
C th thy ngay c hai BT tng ng vi (2) rt quen thuc l
2(a b c) 3(ab bc ca) (3) vi mi a,b,c
2 2 2 2(a b c) 3(a b c ) (4) vi mi a,b,c
Chng ta s ni thm ng dng tuyt vi ca ba BT (2), (3) v (4) nhng phn sau
V d 2: Chng minh rng vi mi a,b,c ta c: 4 4 4a b c abc(a b c)
Gii: p dng lin tip BT (2) trong v d 1 ta c:
4 4 4 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2
a b c (a ) (b ) (c )a b b c c a (ab) (bc) (ac)ab.bc ab.ac bc.ac abc(a b c)
Nh vy nu thi hi cc bn mt bi nh sau:
Cho 3 s thc a,b,c tho mn a b c 1 . Chng minh rng: 4 4 4a b c abc th
chc cc bn c c hi cao t im 10 ri! (Hy c t tin ln nh th!)
V d 3: Chng minh rng vi mi a,b 0 ta c:
3 3 2 2a b a b ab
Gii: Ta bin i 3 3 2 2 2a b a b ab (a b) (a b) 0 , suy ra pcm.
Nhn xt: BT trn tht n gin nhng cng c kh nhiu ng dng vi cc bi ton kh
hn, chng hn ta xt 3 bi ton sau:
Bi 3.1. Cho a,b,c 0 . Chng minh rng:
3 3 3 3 3 31 1 1 1
a b abc b c abc a c abc abc
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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Hng gii: Ta c 3 3 2 2 3 3a b a b ab ab(a b) a b abc ab(a b c)
Suy ra 3 31 1
a b abc ab(a b c)
.
Cng hai BT tng t ta c
1 1 1 1VTab(a b c) bc(a b c) ac(a b c) abc
(pcm).
Xin a ra thm hai h qu ca bi ton trn (coi nh bi tp cho cc bn luyn tp)
* Cho a,b,c 0 tho mn abc=1. Khi : 3 3 3 3 3 31 1 1 1
a b 1 b c 1 a c 1
* Cho a,b,c 0 tho mn abc=1. Khi : 1 1 1 1a b 1 b c 1 a c 1
(che du bn cht hn)
Bi 3.2. Cho a,b,c khng m tho mn a b c 2012 . Tm gi tr nh nht ca biu thc
3 3 3 3 3 33 3 3P 4(a b ) 4(b c ) 4(a c )
Hng gii: Mi nhn BT ta cm thy rt kh khn v c cn bc 3 v iu quan trng l
phi x l c biu thc trong du cn. Bt ng thc 3 3 2 2a b a b ab cho ta mt manh
mi tm ra li gii bi ton, nhng nu p dng nguyn xi nh vy th cha n. Ta bin
i mt cht BT ny
3 3 2 2 3 3 2 2 3 3 3a b a b ab 3(a b ) 3(a b ab ) 4(a b ) (a b)
Nh vy ta c thu c BT 3 3 34(a b ) (a b) .
Chc cc bn cng ng vi ti rng php bin i rt t nhin ch.
By gi p dng BT va tm c ta c
3 3 3 3 3 33 3 3P 4(a b ) 4(b c ) 4(a c ) (a b) (b c) (c a) 2(a b c) 4024
ng thc xy ra khi v ch khi 2012a b c3
.
Vy GTNN ca P bng 4024.
Bi ton tng qut: Cho a,b,c khng m tho mn a b c k . Tm gi tr nh nht ca
3 3 3 3 3 33 3 3P m(a b ) m(b c ) m(a c )
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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( m,k l cc hng s dng cho trc)
Bi 3.3. K hiu A,B,C l ba gc ca mt tam gic bt k. Tm gi tr ln nht ca
3 3 3
3 3 3
sin A sin B sin CPA B Ccos cos cos2 2 2
Hng gii: y qu l mt bi ton kh, ta hy m mm theo cc u mi nh nh
* Th nht: Ta c mt nh gi rt quen thuc trong tam gic:
C A B Csin A sin B 2cos .cos 2cos2 2 2
* Th hai: Cc cn bc 3 gi ta ngh ti BT: 3 33a b 4(a b )
Nh vy, ta c 3 3 3 3 3C Csin A sin B 4(sin A sin B) 4.2cos 2. cos2 2
Tng t ta c 3 3 3 Asin B sin C 2. cos2
v 3 3 3 Bsin A sin C 2. cos2
Cng tng v 3 BT trn ta c
3 3 3 3 3 3A B Csin A sin B sin C cos cos cos2 2 2
Vy P 1 . Du bng xy ra khi v ch khi A=B=C
Do gi tr ln nht ca P bng 1 khi tam gic ABC u.
V d 4: Chng minh rng vi a,b,c l 3 cnh mt tam gic bt k ta c:
2 2 2ab bc ca a b c 2(ab bc ca)
Gii: BT bn tri chng minh, chng minh BT bn phi ta xut pht t mt BT
c bn trong tam gic l b c a b c .
* Nu s dng b c a th ta bin i nh sau:
2 2 2 2 2 2 2a b c a (b c) b c 2bc a b c 2bc
Tng t 2 2 2b a c 2ac ; 2 2 2c a b 2ab . Cng theo tng v ba BT ta c pcm.
* Nu s dng a b c th ta bin i nh sau:
2a b c a ab ac , cng hai BT tng t ta c pcm.
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ATH.
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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V d 5: Chng minh rng vi mi a,b,x, y ta c BT sau (BT Mincpxki)
2 2 2 2 2 2a x b y (a b) (x y) (1)
Gii: Bnh phng hai v v bin i tng ng:
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2
a x b y 2 (a x )(b y ) a x b y 2ab 2xy
(a x )(b y ) ab xy (*)
+ Nu ab xy 0 th hin nhin (*) ng
+ Nu ab xy 0 th 2 2 2 2 2 2(*) (a x )(b y ) (ab xy) (bx ay) 0 (lun ng)
Vy bi ton c chng minh. ng thc xy ra khi bx=ay.
Ch : C th chng minh BT trn bng cch s dng BT vc t rt n gin nh sau
(khi lm bi thi H cc bn phi chng minh BT ny trc khi dng n, lc cc bn
hy chn mt phng n chng minh m cc bn cho l hay v d nh nht. OK).
t u (a; x)
v v (b; y)
, khi u v (a b;x y)
.
T BT vc t u v u v
v cng thc di vc t ta c ngay pcm.
Nu p dng hai ln BT (1) ta c BT sau:
2 2 2 2 2 2 2 2a x b y c z (a b c) (x y z) vi mi a, b,c, x, y,z .
Nhn xt: BT Mincpxki c rt nhiu ng dng hay v c th gii quyt c nhiu bi
BT hc ba. Xin c minh ho iu ny qua 3 bi ton c bn sau y:
Bi 5.1. Cho a,b khng m tho mn a b 1 .
a) Chng minh rng: 2 21 a 1 b 5
b) Tm gi tr nh nht ca 2 22 21 1P a bb a
Hng gii:
a) Ta c 2 2 2 21 a 1 b (1 1) (a b) 5 . ng thc xy ra khi 1a b2
.
b) Ta c 2 2
2 2 2 22 2
1 1 1 1 4P a b (a b) (a b) 17b a a b a b
.
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ATH.
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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ng thc xy ra khi 1a b2
. Vy GTNN ca P bng 17 .
Bi 5.2. Cho x,y,z dng tho mn x y z 1 . Chng minh rng:
2 2 22 2 2
1 1 1x y z 82x y z
Hng gii: p dng BT Mincpxki ta c
22 2 2 2
2 2 2
22
1 1 1 1 1 1P x y z (x y z)x y z x y z
9(x y z) 82 (*)x y z
Du bng xy ra khi v ch khi 1x y z3
.
Vi gi thit x y z 1 ta thay trc tip vo (*) v c kt qu l 82 . Tuy nhin nhiu
khi bi li cho gi thit kh i rt nhiu, mc d du bng vn xy ra khi 1x y z3
.
Chng hn H khi A nm 2003: Cho x,y,z dng tho mn x y z 1 .
Chng minh rng: 2 2 22 2 21 1 1x y z 82x y z
.
Vi bi ton ny ta khng th thay x y z 1 ra ngay kt qu nh bi trn c. ng
trc tnh hung ny ta c ngay hai hng gii quyt.
Hng 1: t 2t (x y z) 0 t 1 . Ta c 81P tt
.
Ta tch kho dng BT Csi: 81 1 80 1 80t t 2 t. 82 P 82t t t t 1
.
Hng 2: Vn t 2t (x y z) v xt hm 81f (t) t ; 0 t 1t
.
Ta c 2
2 2
81 t 81f '(t) 1 0 t 0;1t t
, suy ra hm f(t) nghch bin trn 0;1 .
Do f (t) f (1) 82 P 82 .
Hng gii quyt th hai s c cp phn sau ca chuyn .
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ATH.
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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Bi 5.3. Cho x,y,z dng tho mn x y z 5 . Tm gi tr nh nht ca
2 2 2P 223 x 223 y 223 z
Hng gii: Ta c
2 2 2 2 2 2 2 2P ( 223) x ( 223) y ( 223) z (3 223) (x y z) 2012
ng thc xy ra khi 5x y z3
. Vy GTNN ca P bng 2012 .
C l khng phi ni g thm na th cc bn cng thy v p v sc mnh ca
BT Mincpxki. Nhng ti nhc li rng phi chng minh li BT ny trc khi p
dng nh!
3. Bi tp t luyn
Bi 1. Chng minh rng: , , , , ,a b c d e R ta c:
a) 2 2 2 2 2 ( )a b c d e a b c d e .
b)
33 3
( 0).2 2
a b a b a b
Bi 2. Chng minh rng:
a) 5 5 4 4 2 2( )( ) ( )( ), , : 0.a b a b a b a b a b ab
b) 2 21 1 2 , , 1.
1 1 1a b
a b ab
Bi 3. Cho ABC. Chng minh rng:
a) 2 2 2 3 3 3( ) ( ) ( )a b c b c a c a b a b c .
b) 2 2 2 2( ).a b c ab bc ca
Bi 4. Chng minh rng:
a) (a c)(b d) ab cd , a,b,c,d 0
b) 2 2 2 2 2 2(a c) (b d) a b c d , a,b,c,d R
c) 1 1 1 1 1b (c a) (c a)c a b c a
0 a b c
d) b c a a b ca b c b c a 0 a b c
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ATH.
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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Bi 5. Cho a, b > 0: a + b = 2. Chng minh rng a bab a b
Bi 6. Cho hai s thc a ,b tho mn a + b 2. Chng minh rng : a4 + b4 a3 + b3
Bi 7. Cho ba s a ,b ,c [0;1]. Chng minh rng : a + b + c ab bc ca 1
Bi 8. Cho a,b,c tho mn a b c 1 . Chng minh rng:
a b c a b c
1 1 1 a b c33 3 3 3 3 3
Bi 9. Cho a,b,c dng. Chng minh rng:
2 2 2 2 2 2a ab b b bc c a ac c a b c
II. PHNG PHP S DNG BT NG THC C SI
1. Bt ng thc Csi
a) Cho a 0, b 0 . Khi a b ab2
. ng thc xy ra khi a=b.
b) Cho a 0, b 0, c 0 . Khi 3a b c abc3
. ng thc xy ra khi a=b=c.
Cc dng tng ng l: a b 2 ab ; 2a bab
2
3a b c 3 abc ; 3a b cabc
3
c) Tng qut: Cho n s thc khng m 1 2, ,..., ( 2)na a a n . Khi ta c
1 2 1 2... ... nn na a a n a a a
ng thc xy ra khi v ch khi 1 2 ... na a a .
Ch : Vi cc bi thi H- C thng thng ch cn p dng BT Csi vi 2 hoc 3 s.
2. Mt s v d
V d 1: Chng minh rng:
a) a b 2 a,b 0b a b) a b 2 a,b 0
b a
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ATH.
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Gii. a) p dng BT Csi cho hai s dng ta c: a b a b2 . 2b a b a (pcm)
Du bng xy ra khi a=b.
b) Ta khng th p dng ngay BT Csi v ch c iu kin a,b 0 . Bin i tng
ng BT bng cch bnh phng hai v:
2 2 2
2 2a b a b a b2 4 2b a b a b a
n y theo BT csi th BT sau l ng, vy ta c pcm
Ch l du bng xy ra khi a b .
Cng c th thy ngay rng ab
v ba
cng du nn ta c
a b a b 2b a b a (lc ny li p dng BT Csi c)
V d 2: Cho a,b,c dng. Chng minh rng:
a) 1 1 4a b a b
(1) b) 1 1 1 9
a b c a b c
(2)
Gii. a) Nu vit li BT cn chng minh di dng 1 1(a b) 4a b
th hng gii
quyt l qu r rng. Tht vy, p dng BT Csi cho hai s dng ta c
a b 2 ab v 1 1 12a b ab .
Suy ra 1 1 1(a b) 4 ab. 4a b ab
. Du = xy ra khi v ch khi a=b
b) Hon ton tng t vi phn a) bng cch p dng BT Csi vi 3 s.
Nhn xt: Hai BT trong v d 1 c rt nhiu ng dng v cng l con ng sng to ra
v vn cc BT hay. C th ni phn ln cc BT trong thi H- C c gc tch ca hai
BT ny. Ni ra cc p dng hay ca hai BT ny th nhiu v k v khng bit s tn
km bao giy mc, ti xin ch dn chng ra vi bi ton in hnh.
Bi 2.1. Cho a,b,c dng. Chng minh rng:
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ATH.
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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* Cho x,y,z dng tho mn x 2y 4z 12 . Chng minh rng:
2xy 8yz 4xz 6x 2y 2y 4z 4z x
.
Vi bi ton ny, cc bn ch cn coi a x;b 2y;c 4z th a b c 12 v BT cn
chng minh tr thnh: ab bc ac 6a b b c a c
(y chnh l h qu ca (7) ri. OK)
Bi 2.4. Gi a,b,c l ba cnh ca mt tam gic, p l na chu vi tam gic . Chng minh
rng:
1 1 1 1 1 12p a p b p c a b c
(8)
Hng gii: D thy p a 0;p b 0;p c 0 v nhn xt rng
(p a) (p b) 2p a b c
iu ny gi ta dng BT (1) cho hai s p-a v p-b. C th l:
1 1 4 4p a p b (p a) (p b) c
Cng hai BT tng t ta c BT (8) cn chng minh
Bi 2.5. Cho a,b,c dng. Chng minh rng:
2 2 2 1 1 1 3a b c a b ca b b c c a 2
(9)
Hng gii: Ta c 2 2 2
2 2 2 9 3 3(a b c ) 3VT(9) a b c . . (a b c)2(a b c) 2 a b c 2
( do 2 2 2 2(a b c) 3(a b c ) )
Bi 2.6. Cho x,y,z dng tho mn x y z 1 . Tm gi tr ln nht ca biu thc
x y zPx 1 y 1 z 1
.
Hng gii: c th p dng c BT (2) ta bin i P nh sau:
x 1 1 y 1 1 z 1 1 1 1 1P 3x 1 y 1 z 1 x 1 y 1 z 1
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ATH.
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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Ta c 1 1 1 9 9x 1 y 1 z 1 x y z 3 4
, suy ra 9 3P 34 4
Du bng xy ra khi v ch khi 1x y z3
. Vy GTLN ca P bng 34
.
Vi li gii nh trn cc bn c th lm hon ton tng t vi bi ton tng qut hn
Bi 2.7. Cho x,y,z dng tho mn x y z 1 v k l hng s dng cho trc.
Tm gi tr ln nht ca biu thc x y zPkx 1 ky 1 kz 1
.
Bi 2.8. Cho a,b,c dng tho mn a b c 1 . Tm gi tr nh nht ca biu thc
2 2 21 1 1P
a 2bc b 2ac c 2ab
Hng gii: Ta c ngay 2 2 2 29 9P 9
a 2bc b 2ac c 2ab (a b c)
.
Du bng xy ra khi a b c 1a b ca b c 1 3
. Vy minP 9 .
Bi 2.9. Cho A,B,C l ba gc ca mt tam gic. Chng minh rng:
1 1 1 62 cos2A 2 cos2B 2 cos2C 5
Hng gii: Ta c 1 1 1 92 cos2A 2 cos2B 2 cos2C 6 cos2A cos2B cos2C
D chng minh c rng 3cos2A cos2B cos2C2
(cc bn hy t chng minh nh)
Suy ra 1 1 1 9 632 cos2A 2 cos2B 2 cos2C 562
(pcm)
Bi 2.10. Cho a,b,c dng tho mn a b c 1 . Chng minh rng:
2 2 21 1 1 1 30
ab bc aca b c
(10)
Hng gii: Ta nh gi v tri ca (10) mt cch rt t nhin nh sau:
2 2 2 2 2 21 1 1 1 1 9
ab bc ac ab bc caa b c a b c
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ATH.
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= 2 2 21 1 1 7
ab bc ca ab bc ca ab bc caa b c
29 7 9 7 301ab bc ca 1(a b c)
3
(do BT c bn 21 1ab bc ca (a b c)3 3
)
3. Bi tp t luyn
Bi 1. Cho a, b, c > 0. Chng minh rng:
a) 1 1 1( )( ) 9a b ca b c
. b) 33(1 a)(1 b )(1 c ) 1 abc
c) 2 2 2a b c a b c
b c c a a b 2
d) abc
acb
cbacba
222
222222
Bi 2. Tm gi tr nh nht ca mi biu thc sau:
21
2A aa
vi a > 0. 3 23
B xx
vi x > 0.
Bi 3. Cho a, b, c > 0 tho mn: a + b + c = 2. Tm gi tr nh nht ca: 3 3 3T a b c
Bi 4. Cho x, y, z > 0: x + y + z = 1. Tm Min: 4 4 4R x y z .
Bi 5. Tm gi tr ln nht ca mi biu thc sau:
(3 2 ); (0 3 / 2).M x x x
(1 )(2 )(4 ); (0 x 1, 0 2).N x y x y y
3(1 ) ; 0 1.P x x x
Bi 6. Chng minh rng vi a, b, c > 0 ta c:
a) 6a b b c c ac a b
b) 32
a b cb c c a a b
.
c) bc ca ab a b ca b c
d) 2 2 2 1 1 1 3a b c (a b c)a b b c c a 2
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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e) 2 2 2 2 2 2
a b c 1 1 1 12 a b ca b b c c a
f) 2 2 2 2 2 2
a b b c c a 1 1 1a b ca b b c c a
Bi 7. Cho ABC vi ba cnh l a,b,c. CMR: 3a b cb c a c a b a b c
.
Bi 8. Cho , 1a b . Chng minh rng: 1 1a b b a ab .
Bi 9. Cho ABC. Chng minh rng:
a) 1( )( )( )8
p a p b p c abc .
b) 1 1 1 1 1 12( )p a p b p c a b c
.
Bi 10. Cho a,b,c 0 v a b c 1 . Chng minh rng:
2 2 2
1 1 1 9a 2bc b 2ca c 2ab
Bi 11. Chng minh rng:
a) 1a 3b(a b)
, a b 0 b) 2
1a 2 2b(a b)
, a b 0
c) 2
4a 3(a b)(b 1)
, a b 0 d) 2
2
a 2 2a 1
, a R
e) 2 2
4 4
x y 141 16x 1 16y
x,y R f) 22
1 2(x 1) 1 16
x x
0 x
Bi 12. Cho a,b,c 0 v a b c 1 . CMR: 8abc(a b)(b c)(c a)729
Bi 13. Cho a,b,c 0 v 2 2 2a b c 1 . CMR: 2 2 2 2 2 2
a b c 3 32b c c a a b
Bi 14. Cho a,b,c 0 : 1 1 2a c b . CMR: a b c b 4
2a b 2c b
Bi 15. a) Cho a , b , c > 0 v 1 1 1 21 a 1 b 1 c
. CMR: 18
abc .
b) Cho a,b,c,d 0 tho mn 1 1 1 1 31 a 1 b 1 c 1 d
. CMR: 1abcd81
Bi 16. Cho a, b , c R v a + b + c = 0. CMR: a b c a b c8 8 8 2 2 2 .
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ATH.
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Bi 17. Chng minh rng 2 2( 2) 3 ( 0)2
x xx
Bi 18. Cho a > 0, b > 0 v a + b = 1. Chng minh rng 2 4ab27
.
Bi 19. Chng minh rng nu x > - 3 th
13
93
22
x
x
Bi 20. Chng minh rng nu a > b > 0 th 24a 3
(a b)(b 1)
Bi 21. Tm gi tr nh nht ca biu thc Q = x + y bit x > 0, y > 0 tho mn: 132 yx
Bi 22. Vi xyz = 1, x, y, z > 0. CMR: 23222
yx
zzx
yyz
x
Bi 23. Tm gi tr nh nht ca P = a
bb
a
11
33
vi a, b l cc s dng tho mn iu
kin ab = 1.
Bi 24. Tm gi tr nh nht ca 2 3 Px y
vi x, y l cc s dng tha mn x+y=1.
Bi 25. Cho x, y, z > 0. Chng minh rng
26)(16)(9)(4 z
xyy
zxx
zy
Bi 26. Cho x + y = 1, x, y > 0. Tm gi tr nh nht ca biu thc: xyyx
A 11 22
Bi 27. Cho x,y 0 , x+ y= 1. CMR: 121 22 yx
Bi 28. Cho a + b = 5, a, b > 0. Tm gi tr nh nht ba
P 11
Bi 29. Cho x,y,z dng tho mn xyz=1. Tm gi tr nh nht ca
a) P=x+y+z b) P= x y z c) 1 1 1x y z
Bi 30. Cho 3 s a,b,c > 0. CMR: 2 a
a3 + b2 + 2 b
b3 + c2 + 2 c
c3 + a2 1a2 +
1b2 +
1c2
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ATH.
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Bi 31. Cho x ,y ,z [0;1]. CMR: (2x + 2y + 2z)(2 x + 2 y + 2 z) 818
Bi 32. Cho a 3 ; b 4 ; c 2 . Tm GTLN ca A = ab c 2 + bc a 3 + ca b 4
abc
Bi 33. Cho a,b,c dng . Chng minh rng:
1 1 4 16 64a b c d a b c d
Bi 34. Cho a,b,c dng tho mn a b c 1 . Chng minh rng:
3 3 3 3a b b c a c 18
Bi 35. Cho a,b,c dng tho mn ab bc ca 5 . Chng minh rng:
2 2 23a 3b c 10
Bi 36. Cho a,b,c dng tho mn abc=1. Chng minh rng:
a) 3 3 3a b c a b c
b) 3 3 3 2 2 2a b c a b c
Bi 37. Cho a,b,c dng tho mn abc=1. Chng minh rng:
1 1 1a 1 . b 1 . c 1 1b c a
Bi 38. Gi s a,b,c l ba cnh ca mt tam gic. Chng minh rng:
a b c 3a b c b c a c a b
Bi 39. Cho a,b,c dng tho mn a b c 1 . Chng minh rng:
(1 a)(1 b)(1 c) 8(1 a)(1 b)(1 c)
Bi 40. Cho a,b,c dng tho mn a b c abc . Chng minh rng:
3 3 3a b c 1b c a
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ATH.
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cc bn c thm k thut khi p dng BT Csi ti xin gii thiu mt cht v
phng php chn im ri csi. y c th ni l mt tuyt chiu c o gip cc em
nhanh chng tm ra li gii bi ton.
III. PHNG PHP THM HNG T V CHN IM RI CSI
T vic d on c du bng xy ra (im ri Csi), thm bt cc s hng cho ph
hp v s dng kho lo bt ng thc Csi ta c th t c nhng kt qu khng ng.
c c mt nh hng ng n chng ta thc hin cc bc phn tch bi ton nh
sau:
1. D on du bng xy ra hay cc im m ti t c GTLN, GTNN.
2. T d on du bng, kt hp vi cc BT quen bit, d on cch nh gi (tt
nhin l thm mt cht nhy cm v kh nng ton hc ca mi ngi) cho mi
bi ton. Ch rng mi php nh gi phi m bo nguyn tc du bng xy ra
mi bc ny phi ging nh du bng m ta d on ban u.
lm r iu ny ti xin phn tch cch suy ngh tm ra li gii trong cc v d sau:
V d 1. Chng minh rng vi a,b,c 0 ta c:
2 2 2a b c a b cb c a
Phn tch bi ton:
* Trc ht ta nhn thy nu p dng ngay bt ng thc C si cho 3 s th khng ra
c kt qu mong mun.
* D nhn thy du bng xy ra khi a = b = c.
Khi 2a b
b . V vy ta thm b vo phn t i din
2ab
c chng minh sau:
Li gii.
p dng bt ng thc C si cho hai s dng ta c: 2 2 2
2 2 2 2 2 2
a b cb 2a; c 2b; a 2cb c a
a b c a b cb c a 2a 2b 2c a b cb c a b c a
V d 2. Chng minh rng vi x,y,z > 0 ta c 3 3 3
2 2 2x y z x y zy z x
Phn tch bi ton:
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ATH.
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Ta thy rng vi hng t 3x
y c th c hai hng sau:
Hng 1: Thm 3
2x xy 2xy , cng vi BT c bn 2 2 2x y z xy yz zx
cng cc bt ng thc li ta c iu phi chng minh.
Hng 2: Thm 3 3 3 3 3 3
2 2 2 2 2 2x x y y z zy 3x ; z 3y ; x 3zy y z z x x ri cng li
ta c iu phi chng minh.
V d 3. Cho x, y, z l cc s dng tha mn xyz = 1. Chng minh rng:
x3 + y3 +z3 x + y + z
Phn tch bi ton:
* D on du bng xy ra khi x = y = z = 1.
* Ta mun t hai mc ch l nh gi gim bc t bc 3 xung bc 1 v m bo
du bng khi x=1, nh vy phi s dng BT csi vi 3 s, l iu d hiu. Vy th phi
thm hng s no vo vi 3x . Chc cc bn u thng nht l s 1 ri.
Li gii: p dng BT Csi cho 3 s dng ta c 3x 1 1 3x ; 3y 1 1 3y ; 3z 1 1 3z
Cng tng v 3 BT ta c : 3 3 3x y z 3(x y z) 6
Mt khc 3x y z 3 xyz 3 nn 3(x y z) 6 x y z
Vy bi ton c chng minh.
Cng theo hng ny ta c cc kt qu sau:
Vi x, y, z l cc s dng tha mn xyz = 1 ta c: 3 3 3 2 2 2x y z x y z 2012 2012 2012 2011 2011 2011x y z x y z
(Cc bn hy chng minh cc kt qu ny nh)
V d 4. Cho a, b, c dng tho mn abc=1. Chng minh rng: 3 3 3a b c 3
(1 b)(1 c) (1 c)(1 a) (1 a)(1 b) 4
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ATH.
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Ta s thm cho 3a
(1 b)(1 c) nhng hng t g? tr li c cu hi cc
bn ch l du bng xy ra khi a=b=c=1.
Lc th 3a 1 1 1 1 b 1 c
(1 b)(1 c) 4 8 8 8
V vy ta c cch chng minh sau:
Li gii: p dng BT Csi cho 3 s dng ta c 3 3
3a 1 b 1 c a 1 b 1 c 33. . . a
(1 b)(1 c) 8 8 (1 b)(1 c) 8 8 4
Cng hai BT tng t ta c: 3 3 3a b c 3 1 3(a b c)
(1 b)(1 c) (1 c)(1 a) (1 a)(1 b) 4 2 2
(pcm).
iu phi chng minh.
V d 5. Cho a, b, c dng. Chng minh rng: 3 3 3a b c 1 (a b c)
b(c a) c(a b) a(b c) 2
Phn tch bi ton:
* D on du bng xy ra khi a=b=c.
* Khi 3 3a a a c a b
b(c a) a(a a) 2 4 2
. Vit nh vy v dng ca ta l phi
kh c mu s v tri. Nh vy c th thc hin li gii n gin nh sau:
Li gii: p dng BT Csi cho 3 s dng ta c 3 3
3a c a b a c a b 33. . . a
b(c a) 4 2 b(c a) 4 2 2
Cng hai BT tng t ta c iu phi chng minh.
V d 6. Cho a, b, c dng tho mn a b c 3 . Tm gi tr ln nht ca 3 3 3P a 2010b b 2010c c 2010a
Phn tch bi ton:
* D on P t GTLN ti a b c 1 (tt nhin khng phi lc no iu d on
ca ta cng ng)
* Khi 3 3a 2010b 2011 v d on gi tr ln nht ca P bng 33 2011
(th ny m thi trc nghim th ngon qu...)
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ATH.
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* By gi vi mt tham s m>0 no , ta vit
3 32 23 3
1 1 (a 2010b) m ma 2010b . (a 2010b).m.m .3m m
.
Vn by gi l ta chn m bng bao nhiu th ph hp?
D thy du bng xy ra khi a b 1
m 2011a 2010b m
Li gii: p dng BT Csi cho 3 s dng ta c:
3 32 23 3
1 1 (a 2010b) 2011 2011a 2010b . (a 2010b).2011.2011 .32011 2011
Cng hai BT tng t v cng li ta c:
3
23
1 2011(a b c) 6.2011P . 3. 201132011
Du bng xy ra khi a b c 1 . Vy GTLN ca P bng 33 2011 .
V d 7. Cho a,b,c khng m tha mn a+b+c=3. Tm gi tr nh nht ca
3 3 3P a 64b c .
Phn tch bi ton:
y l bi ton m cc vai tr ca cc bin khng nh nhau. Tuy nhin ta vn d
on c P t GTNN khi a=c. Vn l bng bao nhiu th cha th ni ngay c.
bit iu ta xt hai tham s , 0 v vit P nh sau:
3 3 3 3 3 3 3 3 3 3 3P (a ) (64b ) (c ) 4 2
p dng BT Csi ta c:
2 2 2 3 3P 3 a 3.4 b 3 c 4 2 (*)
Du bng xy ra khi
a c
b / 4 2 3 (1)4
a b c 3
n y vn cha c th tm ra , .
rng gi thit cho a+b+c=3 nn t (*) ta s lm cho cc h s ng trc a,b,c bng
nhau. C th l 2 23 12 (2)
T (1) v (2) d tm ra 24 12,17 17
.
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ATH.
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Khi 24 3a c ,b17 17
v 3
2
12P
17
Mi th th l n. Cc bn hy t vit li li gii bi ton ny v nng nng trong nim
vui chin thng nh.
Nhn xt: Nu ta b gi thit a+b+c=3 th ta c th thu c BT sau:
Cho a,b,c khng m. Chng minh rng
3 3 3 3289(a 64b c ) 64(a b c) .
Li gii ca bi ton ny dnh cho bn c (gi l c th chun ho a v bi
ton trn).
V d 8. Cho x,y,z dng tha mn xy yz xz 1 . Tm gi tr nh nht ca
2 2 2P 3(x y ) z .
Phn tch bi ton:
Chc khng phi bnh lun g th cc bn u cng nhn vi ti rng bi ton ny
qu hay, cu trc p mt nhng khng h d dng. Tt nhin ai chng mong rng bi s
cho tm GTNN ca 2 2 2P x y z hoc vui hn l tm GTNN ca
2 2 2P x y z 2013 .... Ta tr li qu trnh phn tch v tm ti li gii cho bi ton:
iu kin rng buc gi thit l i xng vi x,y,z, nhng trong biu thc P ch
i xng vi x,y; vai tr ca z vi x,y l nh nhau. V vy ta d on P t GTNN khi x=y
v 2z x y
2 (vi 0 no ).
Ta a ra nh gi nh sau: 2
2 zx 2 .xz2 2
;
22 zy 2 .yz
2 2
v 2 2. x y 2. .xy2 2
Do : 2 2 2. x y z 2. . xy yz xz 22 2 2
.
Nh th ta chn 0 sao cho 32
(s 3 trong bi), c th thy ngay
mt s 2 .
Du bng xy ra khi 22 2
1x yxy yz xz 1 z 2x 2y 5z xy yz xz 1 2x y z2 5
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ATH.
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Lc GTNN ca P bng 2.
Cc bn hy bt tay t gii bi ton tng t sau nh:
Cho x,y,z dng tha mn x y z 1 . Tm gi tr nh nht ca
3
3 3 zP x y4
.
IV. PHNG PHP S DNG O HM
1. Ni dung phng php
a) Cc kin thc lin quan:
1. Hm f(x) ng bin trn D khi v ch khi f '(x) 0 x D .
2. Hm f(x) nghch bin trn D khi v ch khi f '(x) 0 x D .
3. Cho hm f(x) ng bin trn D, khi vi u,v D ta c: u v f (u) f (v)
4. Cho hm f(x) nghch bin trn D, khi vi u,v D ta c: u v f (u) f (v)
b) Phng php gii: chng minh BT bng PP o hm, ta kho st s bin thin
ca mt hm s f(x) no c lin quan ti cu trc ca BT cn chng minh. T s bin
thin ca hm s f(x) ta suy ra BT cn chng minh. Ch l cc bin b rng buc theo
gi thit ca bi ton.
cc bn c th hiu ngay t tng ca phng php ny ti xin a ra mt bi
ton n gin sau:
Cho a b . Chng minh rng: 2 21 1a b
a 1 b 1
Cc bn c th chng minh bi ton ny bng PP bin i tng ng, tuy nhin
nhn vo c im hai v ca BT ta xt hm s 21f (x) x
x 1
vi x
Ta c 4 2 4 2 2
2 2 2 2 2 22x x 2x 1 2x x x (x 1)f '(x) 1 0
(x 1) (x 1) (x 1)
vi mi x .
Suy ra hm f(x) ng bin trn .
M a b f (a) f (b) , hay 2 21 1a b
a 1 b 1
(pcm).
Nhn xt: Bi ton trn th hin kh r v PP s dng o hm trong bi ton BT.
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ATH.
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2. Cc dng ton c bn
Trong cc thi vo H- C thng xut hin hai dng bi ton sau:
Dng 1: Bt ng thc cn chng minh ch c mt bin.
V d 1: Chng minh rng vi mi x 0 ta c: xe 1 x (1)
Li gii. Xt hm xf (x) e x 1 vi x 0 . Ta c xf '(x) e 1 0 vi mi x 0 .
Suy ra hm f(x) ng bin trn 0; f (x) f (0) 0 . Vy xe 1 x
Du bng xy ra khi v ch khi x=0.
Nhn xt: Bng vic xt o hm hai ln v s dng v d 1 ta c kt qu sau:
2x xe 1 x
2 vi mi x 0 (2)
Hoc ta c kt qu tng qut hn: Cho n nguyn dng. Chng minh rng:
2 nx x xe 1 x ...
2 n! vi mi x 0 (3)
V d 2: Cho 0 x2
. Chng minh rng:
a) sin x x b) sinx3x
x6
(4) b) sinx 2x
(5)
Li gii. a) Xt hm f(x) x sin x vi x 0;
2. Ta c f '(x) 1 cosx 0 ,
x 0;2
Suy ra hm f(x) ng bin trn 0;
2. Do f(x) f(0) 0 (pcm).
b) Xt hm 3x
f(x) sinx x6
vi x 0;
2. Ta c
2x
f '(x) cosx 12
; f ''(x) sin x x 0 (theo phn a)
Do f '(x) f '(0) 0 f(x) f(0) 0 (pcm).
c) Xt hm
2xf(x) sin x , ta c
2
f '(x) cosx .
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ATH.
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n y kch bn khng nh hai phn (a) v (b) na v f(x) c nghim duy nht
x 0;
2. C l n y cc bn s lng tng. Hy tht bnh tnh v nh li rng khi
o hm f(x) c nghim trong on 0;
2 ta phi ngh ti bng bin thin ca n (hy v
ngay ra nhp BBT i).
T BBT ta suy ra ngay
f(x) f 02
. Vy
2xsinx (pcm).
V d 3: Cho x 2 . Chng minh rng 1 5xx 2
(6)
Li gii. Nu bn no cha tho v vic s dng BT Csi gii bi ton ny th PP s
dng hm s l mt v kh lp l hng . Tht n gin khi ta xt hm s
1
f(x) xx
vi x 2 . Ta c ngay 2
2 2
1 x 1f '(x) 1 0
x x, suy ra hm f(x) ng bin
trn 2; . Do 5f(x) f(2)2
(pcm).
Dng 2: Bt ng thc cn chng minh c nhiu bin
V d 1: Cho a,b,c dng. Chng minh rng:
3
3
a b c abc 10a b c 3abc
(7)
Li gii. t 3
a b ctabc
, khi theo BT Csi th t 3 .
Ta cn chng minh 1 10tt 3
vi t 3 .
n y th bi ton c gii hon ton tng t vi vic chng minh BT (6).
Du bng xy ra khi v ch khi a=b=c.
V d 2: Cho a,b,c dng tho mn 2 2 2a b c 1 . Chng minh rng:
2 2 2 2 2 2
a b c 3 3b c c a a b 2
(8)
Gii. T gi thit ta vit li (8) di dng:
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ATH.
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2 2 2
2 2 22 2 2 2 2 2
a b c 3 3 a b c 3 3 (a b c )1 a 1 b 1 c 2 a(1 a ) b(1 b ) c(1 c ) 2
Ch thm rng a,b,c (0;1)
iu ny cho ta ngh ti vic chng minh 2 2 3x(1 x )9
vi x (0;1)
Xt hm s 2f(x) x(1 x ) trn khong (0;1) ta c ngay kt qu trn.
Ch : C mt cch hi theo dng hnh hc kh th v ca bi ton trn nh sau:
Cho hnh hp ch nht c di ba cnh l a,b,c v di ng cho chnh l 1.
Tm gi tr nh nht ca 2 2 2 2 2 2a b cP
b c c a a b
.
V d 3. (H Khi A-2003): Cho x,y,z dng tho mn x y z 1 . Chng minh rng:
2 2 22 2 2
1 1 1x y z 82
x y z (9)
Gii. t v tri ca (9) l P. Theo cc kt qu quen bit ta c:
2 2
2 21 1 1 9P (x y z) (x y z)x y z x y z
Nu gi thit cho x y z 1 th qu n ri, nhng y gi thit li cho x y z 1 .
Gi thit ny thng lm cho cc em HS bi ri. Ti sao khng ngh ti hm s nh. Cc
bn ch cn t 2(x y z) t th 0 t 1 v BT cn chng minh tr thnh 81t 82t
vi 0 t 1 . iu ny th qu n gin ri.
Ch l du bng vn xy ra khi x y z 1 v x=y=z, hay 1x y z3
.
Nh vy sc mnh ca PP hm s tht kinh khng v c th xuyn thng bt k
hng phng ng no cho d gi thit v kt lun ca bi ton mi nhn chong. Ti
xin dn chng thm mt vi bi BT khng i xng na, c l cc bn s tm phc
khu phc iu ti ni thi.
V d 4. Cho a,b,c dng tha mn a+b+c=1. Tm gi tr nh nht ca
3 3 31P a b c4
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Gii. S dng nh gi 3
3 3 (a b)a b4
, suy ra
3 33 3 2(a b) 1 (1 c) 1 1P c c (3c 3c 1)
4 4 4 4 4
Ta thnh cng khi a BT vi 3 bin v ch cn 1 bin c, ch l c (0;1) .
Xt hm 21f (c) (3c 3c 1)4
, ta c 1 1f '(c) (6c 3) 0 c4 2
T 1 1f (c) f2 16
. Du bng xy ra khi
1a b a b4
1 1c c2 2
.
Vy GTNN ca P bng 116
, khi 1 1a b ;c4 2
.
V d 5. Cho a,b,c khng m tha mn iu kin a+b+c=1. Chng minh rng:
7ab bc ca 2abc27
(10)
Gii. W.L.O.G, gi s c l s nh nht trong 3 s a,b,c. Khi 1c 0;3
.
Ta s tp trung vo vic nh gi VT ca (10) theo bin c.
Ta c 2a bab bc ca 2abc c(1 c) ab(1 2c) c(1 c) (1 2c)
2
= 21 cc(1 c) (1 2c)
2
.
Xt hm s 21 cf (c) c(1 c) (1 2c)
2
vi 1c 0;
3
Ta c 1f '(c) c(1 3c) 02
, suy ra hm f(c) ng bin trn 10;3
Do 1 7f (c) f3 27
(pcm). Du bng xy ra khi v ch khi 1a b c3
.
3. Bi tp t luyn
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Bi 1. Chng minh rng:
a) , 0x sinx x b) 2
1 , 02
x xe x x
c) tan 2 , 0;2
sinx x x x
Bi 2. Tm m : 2 4 22 0 m x x m x .
Bi 3. a) Cho x (0;1) . CMR: 2 2 3x(1 x )9
.
b) vi 2 2 2, , 0 : 1a b c a b c ta c: 2 2 2 2 2 23 3
2
a b c
b c c a a b
Bi 4. Cho ab0. CMR: 2
1a
a 1
2
1b
b 1
.
Bi 5. Cho >1. CMR vi mi x0 th : (1+x) 1+x.
Bi 6. a) CMR vi mi x th 5 5 1x (1 x)16
.
b) Tng qut: n nn 1
1a b
2 vi n nguyn dng; a,b>0 tho mn a+b=1
c) Tm GTNN ca f(x)= 10 10sin x cos x .
Bi 7. Chng minh rng vi mi x>0 th: lnx< x .
Bi 8. Cho 0 x2
. CMR:
3 12sin tan 22 2 2x
x x .
Bi 9. Cho 00. CMR: x y x y2 ln x ln y
.
Bi 11. Chng minh rng 2 2x x 1 x x 3 1 2 .
Bi 12. (Khi A-2003). Cho x,y,z>0 tho mn: x+y+z1. Chng minh rng:
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2 2 22 2 2
1 1 1x y z 82
x y z .
Bi 13. (Khi D-2001). CMR vi mi x0 v vi mi >1 ta lun c x 1 x .
T hy CMR: Vi ba s dng a,b,c bt k ta c:
3 3 3
3 3 3
a b c a b c
b c a b c a .
Bi 14. Chng minh rng vi mi x0 ta c: cosx2x
12
.
Bi15. Cho tam giac ABC nhn. Chng minh rng
sin A sin B sinC tan A tan B tanC 2
Bi 16. Chng minh rng mi x 0 th ta c: x x 2e e 2 ln(x 1 x ) .
Bi 17. Cho 2
cosxx 0; . CMR: 8
4 sin x(cosx-sinx).
Bi 18. Tm GTNN ca hm s 29
f(x) 4x sin x, x 0x
Bi 19. Cho 430 . Chng minh rng 31.2 2
Bi 20. Cho x y z 0 . Chng minh rng:
x z y x y zz y x y z x
Bi 21. Cho x,y,z khng m tha mn 2 2 2x y z 1 . Chng minh rng:
6(y z x) 27xyz 10
Bi 22. Cho x,y,z, dng tho mn x+y+z=4 v xyz=2. Tm GTLN v GTNN ca
4 4 4P x y z
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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V. MT S THI I HC V LI GII
Bi gii
Vi a, b, c > 0 ta chng minh c: 3 3 2 2
3 3 2 2
3 3 2 2
a b a b abb c b c bcc a c a ca
VT (1) 2 2 2 2 2 2
1 1 1a b ab abc b c bc abc c a ca abc
= 1 1 1ab(a b c) bc(a b c) ca(a b c)
= 1 1 1 1 1a b c ab bc ca abc
.
Bi gii
* Ta c: 21 x 2x 2
x 121 x
. Tng t 2
y 121 y
v 2
z 121 z
Suy ra 2 2 2
x y z 321 x 1 y 1 z
* Ta c: 1 1 1 9 9 9 31 1 1 (1 ) (1 ) (1 ) 3 ( ) 6 2
x y z x y z x y z
Bi gii
Bi 2: Cho x, y, z 0 tho mn x y z 3 . Chng minh rng:
2 2 2
x y z 3 1 1 12 1 x 1 y 1 z1 x 1 y 1 z
Bi 3: Cho a, b, c > 0 sao cho abc = 1. Tm GTNN ca 2 2 2 2 2 2
bc ca abPa b a c b a b c c a c b
Bi 1: Chng minh rng vi a,b,c 0 ta c
3 3 3 3 3 3
1 1 1 1abca b abc b c abc c a abc
(1)
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Bin i 2 2 2
bc ca abPa (b c) b (a c) c (a b)
=2 2 2
1 1 1a b c
b c a c a bbc ac ab
= 2 2 2
1 1 1a b c
1 1 1 1 1 1b c c a a b
n y th mi th tr ln nh nhng hn nhiu ri
Ta ch cn t 1xa
, 1yb
, 1zc
, khi xyz = 1 v 2 2 2x y zP
y z z x x y
p dng BT Csi ta c: 2x y z x
y z 4
, 2y z x y
z x 4
, 2z x y z
x y 4
x y zP x y z2
3x y z 3 3P xyz
2 2 2
. Vy min
3P2
khi x = y = z = 1
Bi gii
Ta c (1) tng ng
3 3 3 3 3 3 3 3 3 2 2 2 3 2 3 2 2 3a b c a b b c c a ab bc ca a b c ab c a bc .
p dng BT csi cho ba s dng ta c:
3 3 3 2a b b 3ab
3 3 3 2b c c 3bc
3 3 3 2c a a 3ca
T suy ra 3 3 3 2 2 23(a b c ) 3(ab bc ca ) 3 3 3 2 2 2a b c ab bc ca (2)
Tng t ta c 3 3 3 3 3 3 3 2 2 3 3 2a b b c c a ab c a bc a b c (3)
T (2) v (3) suy ra: 3 3 3 3 3 3 3 3 3 2 2 2 3 2 3 2 2 3a b c a b b c c a ab bc ca a b c ab c a bc
Bi 4: Cho a, b, c > 0. Chng minh rng:
3 3 3 2 2 21 a 1 b 1 c 1 ab 1 bc 1 ca (1)
Bi 5: Cho a, b, c > 0: 1 1 1 3a b c . Chng minh rng 1 a 1 b 1 c 8 (1)
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Bi gii
T 1 1 1 3a b c suy ra 3 2 2 23abc ab bc ca 3 a b c abc 1
Bin i (1) 1 1 1 81 1 1 0a b c abc
(2)
Ta c VT(2) = 1 1 1 1 1 1 71a b c ab bc ca abc
= 1 1 1 74ab bc ca abc
3 2 2 2
1 74 3abca b c
3 74
abc abc (do abc 1 )
44 0abc
(do abc 1 ) (pcm).
Bi gii
Ta c: 3 2 3 2 3 2 3 2 3 2 3 2
2 y 2 y2 x 2 z 2 x 2 zx y y z z x 2 x y 2 y z 2 z x
=
= 1 1 1 1 1 1x y y z z x
2 2 2 2 2 2
1 1 1 1 1 1 1 1 12 2 2x y y z z x
= 2 2 2
1 1 1x y z
Bi gii
Theo BT Csi ta c 1 1x y 4x y
, x,y 0 v
2
1 4xy (x y)
, x,y 0
Do 2 2 2 2
1 1 1 1 1ab 2ab 2aba b a b
2 2 2
4 4 2 4 62(a b) 2ab a b
.
Bi 6: Cho x, y, z > 0. Chng minh rng:
3 2 3 2 3 2 2 2 2
2 y2 x 2 z 1 1 1x y y z z x x y z
Bi 7: Cho a, b > 0 tho mn a + b = 1. Chng minh rng
2 2
1 1 6ab a b
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Bi gii
Gp nhng bi ton dng ny ta thng s dng BT sau (BT Mincpxki)
22 2 2 2 2( ) x a y b x y a b (1) vi mi a,b,x,y
22 2 2 2 2 2 2( ) x a y b z c x y z a b c (2) vi mi a,b,c,x,y,z
Chng minh BT (1) tht n gin, c th a ra 2 cch nh sau:
Cch 1: Bin i tng ng (1) bng cch bnh phng hai v v a v BT ng 2( ) 0 bx ay
Cch 2: S dng BT vc t
Vi mi vc t ,
u v ta c
u v u v (*)
(v 22 2 22 2
2 . 2 .
u v u v u v u v u v u v )
t ( ; )
u x a , ( ; )
v y b . p dng (*) suy ra ngay BT (1)
p dng hai ln BT (1) ta suy ra BT (2), v cng suy ra cc BT tng qut cho b 2n
s
Tr li bi ton: p dng BT (2) ta c:
2
2 2 2 22 2 2
1 1 1 1 1 1P x y z (x y z)x y zx y z
22
3 313 xyz 3
xyz
99tt
Vi 23 ( )t xyz2
x y z 10 t3 9
.
n y th tng dng hm s l kh r rng ri
Xt hm s 9f (t ) 9tt
vi 1t 0,9
Bi 8: (Khi A- 2003). Cho x, y, z l ba s dng tho mn x y z 1 . Chng minh rng:
2 2 22 2 2
1 1 1x y z 82x y z
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Ta c /2
9 1f (t ) 9 0, t 0,9t
f (t ) gim trn 10,
9
1f (t ) f 829
.
Vy P f(t ) 82 .
Bi gii
T gi thit v BT cn chng minh cho php ta ngh ngay ti BT c bn sau:
* Vi a,b dng ta c 1 1 4a b a b
, hay 1 1 1 1
a b 4 a b
.
Du = xy ra khi v ch khi a = b
* p dng kho lo BT trn ta c
1 1 1 1 1 1 1 1 1 1 1 1 12x y z 4 2x y z 4 2x 4 y z 8 x 2y 2z
(1)
1 1 1 1 1 1 1 1 1 1 1 1 1x 2y z 4 2y x z 4 2y 4 x z 8 y 2x 2z
(2)
1 1 1 1 1 1 1 1 1 1 1 1 1x y 2z 4 2z y x 4 2z 4 y x 8 z 2y 2x
(3)
Cng theo tng v cc BT (1), (2) v (3) ta c
1 1 1 1 1 1 1 12x y z x 2y z x y 2z 4 x y z
Bi gii
* Mu cht ca bi ton l phi bit phn tch cc c s v tri
212 15. 35 4
; 212 20. 45 3
; 215 20. 54 3
* n y p dng BT Csi cho hai s dng ta c
Bi 9: (Khi A 2005). Cho x, y, z l cc s thc dng tho mn iu kin 1 1 1 4x y z .
Chng minh rng: 1 1 1 12x y z x 2y z x y 2z
Bi 10: (Khi B 2005). Chng minh rng vi mi x R , ta c x x x
x x x12 15 20 3 4 55 4 3
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12 15 12 152 . 2.35 4 5 4
x x x xx (1)
12 20 12 20. 2.45 3 5 3
x x x xx (2)
15 20 15 202 . 2.54 3 4 3
x x x xx (3)
Cng cc BT (1), (2), (3) ta c x x x
x x x12 15 20 3 4 55 4 3
(pcm).
Du = xy ra khi v ch khi x=0.
Bi gii
* p dng BT C si cho ba s dng ta c
3 3 3 331 x y 3 1.x .y 3xy 3 31 x y 3
xy xy
(1)
* Cng hai BT tng t ta c
3 3 3 3 3 31 x y 1 y z 1 z x 3 3 3xy yz zx xy yz zx
* Mt khc: 33 3 3 3 3 33 3 3xy yz zx xy yz zx
3 3 3 3 3 31 x y 1 y z 1 z x 3 3
xy yz zx
Bi gii
Bi 11: (Khi D 2005). Cho cc s thc dng x, y, z tho mn xyz = 1.
Chng minh rng: 3 3 3 3 3 31 x y 1 y z 1 z x 3 3
xy yz zx
Bi 12: (D b D 2005). Cho x, y, z l ba s thc tho mn x y z 0 . Chng minh rng
x y z3 4 3 4 3 4 6
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* Bi ton ny c cht gn gi vi bi 11, nu tinh ta thy rng
x y z x y z 04 .4 .4 4 4 1
* Ta c : 4x x x3 4 1 1 1 4 4 4 4 8x x x3 4 2 4 2 4
* Cng hai BT tng t ta c
38 8 8x y z x y z x y z83 4 3 4 3 4 2 4 4 4 6 4 .4 .4 6
Du = xy ra khi v ch khi x=y=z=0.
Bi gii
* Ta c 3
43
x x x x1 x 1 43 3 3 3
Tng t : 3
43 3
y y y y y1 1 4x 3x 3x 3x 3 .x
V
3
4 3
9 3 3 3 31 1 4y y y y y
26
43
9 31 16yy
Do 2
3 3 6
43 3 3 3
y 9 x y 3(1 x) 1 1 256 256x 3 3 x yy
Bi gii
Cch 1
Ta c
3 3
a 3b 1 1 1a 3b (a 3b)1.1 (a 3b 2)3 3
Bi 13: (D b A 2005). Chng minh rng vi mi x,y 0 , ta c
2y 9(1 x) 1 1 256x y
Bi 14: (D b B 2005). Cho a, b, c l ba s dng tho mn 3a b c4
.
Chng minh rng
3 3 3a 3b b 3c c 3a 3
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3 3
b 3c 1 1 1b 3c (b 3c)1.1 (b 3c 2)3 3
3 3
c 3a 1 1 1c 3a (c 3a)1.1 (c 3a 2)3 3
3 3 3 1 1 3a 3b b 3c c 3a 4(a b c) 6 4 6 3
3 3 4
Du = xy ra
3a b c 1a b c44a 3b b 3c c 3a
Cch 2
t
3x a 3b 3x a 3b
3y b 3c 3y b 3c
3z c 3a 3z c 3a
3 3 33x y z 4(a b c) 4 34
. Ta cn chng minh x y z 3
BT ny l qu n gin ri! ( cc em t chng minh xem nh).
Bi gii
T 0 x 1 suy ra 2x x
p dng BT C si cho ba s dng ta c
2 21 1 1y x yx 2 yx x y4 4 4
1x y y x4
(pcm).
Du bng xy ra
2
2
0 x y 1
x x1yx4
x 11y4
Bi 15: (D b 2 B 2005). Chng minh rng nu 0 x y 1 ta c
1x y y x4
.
Bi 16: (D b 2 D 2005). Cho x, y, z dng tho mn x.y.z 1 . Chng minh rng
2 2 2x y z 31 y 1 z 1 x 2
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Bi gii
Ta c
2 2x 1 y x 1 y2 x1 y 4 1 y 4
Cng hai BT tng t ta c
2 2 2x 1 y y 1 z z 1 x x y z
1 y 4 1 z 4 1 x 4
2 2 23x y z 3(x y z) 3 3 3 9 3 33. xyz
1 y 1 z 1 x 4 4 4 4 4 4 2 (pcm).
Bi gii
t xa 3 , yb 3 , zc 3 (a, b, c 0)
Ta c : x y z3 3 3 1 1 1 1 1a b c
ab bc ca abc (*)
BT cn chng minh tr thnh:
2 2 2a b c a b ca bc b ca c ab 4
3 3 3
2 2 2
a b c a b c4a abc b abc c abc
(2)
Ta c: 2 2a abc a ab bc ca (a b)(a c) (do ab bc ca abc )
Tng t 2b abc (b a)(b c) v 2c abc (c a)(c b)
Do (2)
3 3 3a b c a b c(a b)(a c) (b a)(b c) (c a)(c b) 4
(3)
p dng BT C si cho ba s dng ta c
3 33
a a b a c a 33. a(a b)(a c) 8 8 64 4
Tng t
3 33
b b c b a b 33. b(b c)(b a) 8 8 64 4
3 33
c c a c b c 33. c(c a)(c b) 8 8 64 4
Bi 17: (D b A 2006). Cho x,y,z dng tho mn x y z3 3 3 1 . Chng minh rng:
x y z x y z
x y z y z x z x y
9 9 9 3 3 343 3 3 3 3 3
(1)
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Cng tng v ca ba BT ta c:
3 3 3a b c 1 3(a b c)(a b c)(a b)(a c) (b a)(b c) (c a)(c b) 2 4
3 3 3a b c a b c(a b)(a c) (b a)(b c) (c a)(c b) 4
, suy ra (3) c chng minh
Bi ton c chng minh hon ton.
Bi gii
* Gi thit xyz=1 thng c rt nhiu k thut x l tinh t, vic dng n th no
cn ph thuc vo tng bi ton. Trong bi ton ny c mt lin h nh gia t v mu, c
th ta thc hin li gii nh sau:
Ta c 2 2 2 4x (y z) x y x z 2 x yz 2x x (do xyz=1)
Tng t 2y (z x) 2y y v 2z (x y) 2z z
2y y2x x 2z zPy y 2z z z z 2x x x x 2y y
* n y t v mu qu gn gi ri. bi ton n gin hn ta thc hin
php i bin khng lm thay i gi thit sau: a x x;b y y;c z z , khi abc=1.
Lc 2a 2b 2cPb 2c c 2a a 2b
* Cc bn hy chng minh BT sau nh (rt quan trng y, qu nhiu p dng hay)
a b c 3mb nc mc na ma nb m n
(vi mi a,b,c dng; m,n l hng s dng cho trc)
* p dng BT ny ta c ngay P 2 (cng chng cn n abc=1 na)
Du bng xy ra khi x=y=z=1. Vy GTNN ca P bng 2.
Bi 18: (Khi A 2007). Cho x, y, z l cc s thc dng tho mn xyz=1.
Tm gi tr nh nht ca 2 2 2( ) ( ) ( )
2 2 2
x y z y z x z x yPy y z z z z x x x x y y
Bi 19: (Khi B 2007). Cho x, y, z l cc s dng thay i. Tm gi tr nh nht ca
biu thc 1 1 12 2 2
x y zP x y zyz zx xy
.
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Bi gii
Ta c 2 2 2 2 2 2x y z x y zP
2 2 2 xyz
M 2 2 2 2 2 2
2 2 2 x y y z z xx y z xy yz zx2 2 2
2 2 2x y z xy yz zx 1 1 1
xyz xyz x y z
2 2 2 2 2 2x y z 1 1 1 x 1 y 1 z 1P2 2 2 x y z 2 x 2 y 2 z
tng dng hm s li n rt t nhin
Xt hm s 2t 1f (t )
2 t vi t >0 / 2
1( ) 0 1f t t tt
Lp bng bin thin ta c 3f (t )2
, t 0 . Suy ra 9P2
.
Du bng xy ra khi v ch khi x y z 1 . Vy GTNN ca P bng 92
.
Bi gii
Bt ng thc cho tng ng vi
a b
b aa b ln 1 4 ln 1 41 4 1 4a b
Xt hm s xln 1 4
f (x)x
vi x>0. Ta c: x x x x
2 x
4 ln 4 (1 4 ) ln 1 4f '(x) 0
x (1 4 )
Suy ra f(x) nghch bin trn (0; ) .
Bi 20: (Khi D- 2007). Cho a b 0. Chng minh rng b a
a ba b
1 12 22 2
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Do f(x) nghch bin trn (0; ) v a b 0 nn f (a) f (b) v ta c iu phi chng
minh.
Bi gii
Ta c 2 2
2 2 2 22(x 6xy) 2(x 6xy)P
1 2xy 2y x y 2xy 2y
* Nu y=0 th 2x 1 P 2 .
* Nu y 0 th
2
2
2 2
x x2 6y y 2t 12tP
t 2t 3x x2 3y y
n y c th dng iu kin c nghim ca phng trnh bc hai hoc dng o
hm bng cch kho st hm s 2
22t 12tf (t)t 2t 3
, vi t .
Ta tm ra max3 1P 3 x , y10 10
hoc 3 1x , y10 10
min3 2P 6 x , y13 13
hoc 3 2x , y13 13
Bi gii
Ta s p dng BT sau: 2(a b) 4ab
Ta c 22 2
(x y)(1 xy) (x y)(1 xy) 1P(1 x) (1 y) 4(x y) (1 xy)
. Suy ra 1 1P
4 4
Bi 21: (Khi B- 2008). Cho hai s thc x,y thay i tho mn h thc 2 2x y 1 .
Tm GTLN, GTNN ca 2
22(x 6xy)P
1 2xy 2y
.
Bi 22: Khi D- 2008). Cho hai s thc khng m x,y thay i . Tm GTLN, GTNN ca
2 2(x y)(1 xy)P(1 x) (1 y)
.
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Khi x=0, y=1 th 1P4
. Khi x=1, y=0 th 1P4
.
Vy GTNN ca P bng 14
, GTLN ca P bng 14
.
Bi gii
Ta c 3 3 34(a b ) (a b) a,b 0 nn 3 33 4(a b ) a b
p dng BT ny ta c
2 2 2 2 2 2
x y z x y zP (x y) (y z) (x z) 2 2 x y zy z x y z x
n y ta dng BT csi cho 6 s ta c
62 2 2
x y zP 2.6 x.y.z. . . 12y z x
ng thc xy ra khi x=y=z=1. Vy GTNN ca P bng 12.
Bi gii
Ta c 3 33x 4(y z ) x y z , suy ra 3 33
y z y zx y zx 4(y z )
Cng hai BT tng t ta c pcm.
Bi 23 (D b A- 2007): Cho x,y,z dng. Tm gi tr nh nht ca biu thc
3 3 3 3 3 33 3 32 2 2
x y zP 4(x y ) 4(y z ) 4(z x ) 2y z x
Bi 24: Cho x,y,z dng. Chng minh rng
3 3 3 3 3 33 3 3
y z x z x yP 2x 4(y z ) y 4(x z ) z 4(x y )
Bi 25: Cho a,b,c dng tho mn iu kin 3 1a b c 12 abc
. Tm gi tr nh nht ca
3 3 3
2 2 2
a b b c c aP1 ab 1 bc 1 ca
.
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Bi gii
Bin i 2 2 2 2a b c (a b c)P 1 1 1 a b cb c a (a b c)
ab bc ca abc
= 2(a b c)
1(a b c) 1abc
Theo gi thit ta c 1 21 (a b c)abc 3
, suy ra 2(a b c) 3P 2 2(a b c). (a b c)
3
Du bng xy ra khi
4 3 3 2a b c
2a a 1 0 (a 1)(2a a a 1) 0 a 11 21 (a b c)abc 3
Vy GTNN ca P bng 32
khi a b c 1 .
Bi gii
Ci hay v kh ca bi ton l gi thit (x + y)3 + 4xy 2 (1)
Biu thc A ch cha cc s hng x2 + y2 v x2y2. iu ny cho ta ngh ti vic nh gi
tng x+y v qua nh gi c tng bnh phng x2 + y2 .
C th ta bin i gi thit (1) nh sau:
Ta c 3 2 3 3 2(x y) (x y) (x y) 4xy 2 (x y) (x y) 2 0 x y 1 .
Suy ra 2
2 2 (x y) 1x y2 2
. Ta bin i A nh sau:
2 2 22 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2
(x y )A 3 (x y ) x y 2(x y ) 1 3 (x y ) 2(x y ) 14
9 (x y ) 2(x y ) 14
t t = x2 + y2 th t 12
. Xt hm 29 1f (t) t 2t 1, t4 2
ta tm ra 9min f (t)16
khi 1t2
.
Bi 26 (H KB-2009). Cho cc s thc x, y thay i v tho mn (x + y)3 + 4xy 2. Tm gi tr nh nht ca biu thc :
A = 3(x4 + y4 + x2y2) 2(x2 + y2) + 1.
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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Vy min9 1A khi x y
16 2 .
Bi gii
Trc ht ta bin i P theo tch xy. Ta c
P= (4x2 + 3y)(4y2 + 3x) + 25xy = 16x2y2 + 12(x3 + y3) + 34xy
= 16x2y2 + 12[(x + y)3 3xy(x + y)] + 34xy = 16x2y2 + 12(1 3xy) + 34xy = 16x2y2 2xy + 12
t t = x.y, v x, y 0 v x + y = 1 nn 104
t . Khi S = 16t2 2t + 12
Xt hm f(t)=16t2 2t + 12 vi 1t 0;4
, ta tm ra min, max ca hm f(t)
p s ca bi ton l: Max P = 252
khi x = y = 12
Min P = 19116
khi 2 3x
42 3y
4
hay 2 3x
42 3y
4
Bi gii
T gi thit ta c (x+y)(x+z)=4yz.
Do BT tng ng vi 3 3 3(x y) (x z) 12yz(y z) 5(y z) (1)
Mt khc
22 2 2
3 3
3(y z)x (y z)x 3yz 4x 4(y z)x 3(y z) 04
(2x y z) 2x 3(y z) 0 2x y z 8x (y z)
p dng BT c bn: 3 3 3(x y) 4(x y ) ta c 3 3 3 3 3 3 3VT(1) 4(x y ) 4(x z ) 12yz(y z) 8x 4(y z) 5(y z) (pcm)
Du ng thc xy ra khi v ch khi x=y=z.
Bi 27 (H KD-2009). Cho cc s thc khng m x, y thay i v tha mn x + y = 1. Tm GTLN, GTNN ca P = (4x2 + 3y)(4y2 + 3x) + 25xy.
Bi 28 (H KA-2009): Chng minh rng vi mi s thc dng x, y, z tha mn x(x + y + z) = 3yz, ta c
3 3 3x y x z 3 x y x z y z 5 y z .
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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Bi gii Ta bin i P theo mt hm s ca bin t=ab+bc+ca. Ta c BT c bn 2 2 2 2 2 2 2 23( ) ( )a b b c c a ab bc ca t Suy ra 2 3 2 1 2P t t t .
Mt khc 2( ) 3( )a b c ab bc ca suy ra 103
t .
Xt hm 2( ) 3 2 1 2f t t t t vi 103
t .
Ta c 2 (2 3) 1 2 2'( ) 2 3
1 2 1 2t tf t t
t t
; '( ) 0 (2 3) 1 2 2f t t t (1)
t 1 2u t th 1 13
u , pt (1) tr thnh: 3 4 2 0u u (2)
D chng minh c 3 4 2 0u u vi mi 1 13
u , do (2) v nghim, tc l (1) v
nghim. T suy ra ngay '( ) 0f t vi mi 103
t .
Do ( ) (0) 2f t f . Vy minP=2 khi chng hn a=b=0 v c=1.
Bi gii Tp xc nh ca hm s 2;5D .
Ta c 2 2
2 4 2 3'( )2 4 21 2 3 10
x xf xx x x x
, vi ( 2;5)x .
2 2'( ) 0 ( 2 4) 3 10 ( 2 3) 4 21f x x x x x x x Suy ra 2 2 2 2( 2 4) ( 3 10) ( 2 3) ( 4 21)x x x x x x (1)
Khai trin ta c 2815151 104 29 013
xx x
x
.
Th li ch c 13
x l nghim ca (1).
Lp bng bin thin ca hm s f(x) suy ra 1 200 98minf(x)3 3
f
.
Cc thi nm 2011, 2012, 2013 cc bn c th tham kho -p n ca b gio dc ti
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Bi 29 (H KB-2010). Cho a,b,c khng m tha mn a+b+c=1. Tm GTNN ca: 2 2 2 2 2 2 2 2 23( ) 3( ) 2P a b b c c a ab bc ca a b c
Bi 30 (H KD-2010). Tm gi tr nh nht ca hm s 2 2( ) 4 21 3 10f x x x x x
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
46
Trn y ch l mt cch tip cn vi phng php v cch hc v BT m ti mun
gii thiu vi cc bn, c bit l cc em hc sinh lp 12 chun b thi H- C. Cn mt s
chuyn su hn ti s tip tc gi ti cc bn sau. Hy vng cht t kin thc ny s gip
cc em hc sinh 12 t c kt qu cao nht trong k thi H sp ti. Chc cc em thnh
cng. Mi thc mc hay chia s hy lin h vi ti theo a ch
Email: [email protected]; ST: 0912960417
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MT S BI TP CHO CC BN LUYN TP
1. Cho a,b,c dng tha mn a+b+c=1. Tm GTNN ca 3 3 31P a b c4
2. Cho a,b,c khng m tha mn: a+b+c=1. Chng minh rng:
2ab bc ca 2abc27
3. Cho x y z 0 . Chng minh rng: x z y x y zz y x y z x
4. Cho a>b>c>0. Chng minh rng: 3 2 3 2 3 2 2 3 2 3 2 3a b b c c a a b b c c a
5. Cho x,y,z dng. Chng minh rng: 4 4 4 2 2 2 2 2 2(x y z ) xyz(x y z) xy(x y ) yz(y z ) xz(x z )
6. Cho x,y,z khng m tha mn 2 2 2x y z 1 . Chng minh rng:
6(y z x) 27xyz 10
7. Cho a,b,c dng v k>0 cho trc. Tm GTNN ca
a b b c a c kabcPc a b (a b)(b c)(a c)
8. Cho x,y,z dng tha mn 3(x y z) 32xyz . Tm GTLN, GTNN ca 4 4 4
4
x y zP(x y z)
.
9. Cho dy 4 3 2nu n 20n 0,5n 13n n 1 . Tm s hng ln nht ca dy v tnh s
hng y.
10. Cho x,y l cc s t nhin. Tm GTLN ca hm s
1 1f (x, y)x y 1 (x 1)(y 1)
11. Cho a, b, c l ba s thc dng. Chng minh bt ng thc: 2a b c
b c c a a b
.
12. Cho ba s dng a, b, c tha a + b + c 2. Chng minh :
2 2 21 1 1 1
a bc b ca c ab abc
13. Cho ba s dng a, b, c tha: 1 1 1 2a b c . Chng minh bt ng thc:
1 1 1 13 3 3a b b c c a
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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14. Xt cc s thc dng x, y, z tha mn iu kin x + y + z = 1.
Tm gi tr nh nht ca biu thc: 2 2 2x (y z) y (z x) z (x y)P
yz zx xz
15. Tm gi tr ln nht v gi tr nh nht ca hm s 4 2 2
2 2
2 1 1 11 1 2x x xy
x x
16. Tm gi tr ln nht v gi tr nh nht ca hm s 21 1y x x
17. Cho x, y, z l cc s thc thuc on [0; 1] v tha mn x + y + z = 1. Tm gi tr ln nht ca
biu thc P = x2 + y2 + z2.
18. Cho x, y, z l cc s thc tha mn x2 + y2 + z2 = 1. Tm gi tr ln nht, nh nht ca biu thc:
P = x3 + y3 + z3 3xyz.
19. Cho hai s thc ,x y thay i v tho mn iu kin 2 2 11x y . Hy tm gi tr ln nht v gi
tr nh nht ca biu thc 2P x xy .
20. Cho hai s thc x, y thay i v tho mn 2 2 8x y . Hy tm gi tr ln nht v gi tr nh
nht ca biu thc 3 3 3P x y xy .
21. Cho x, y l hai s thc thay i v tho mn iu kin: 2 2x y x y . Hy tm gi tr ln nht
v gi tr nh nht ca biu thc 3 3A x y .
22. Cho ,x y l hai s thc thay i v tho mn iu kin: 2 2 2 2 2x y x y . Hy tm gi tr
ln nht v gi tr nh nht ca biu thc 2 2A x y .
23. Cho ba s thc dng x, y, z. Tm gi tr nh nht ca biu thc:
x y z y z z x x yPy z z x x y x y z
.
24. Cho cc s thc x, y, z thuc khong (0; 1) v tha mn: xy + yz + zx = 1. Tm gi tr nh nht
ca biu thc: 2 2 2(1 )(1 )(1 )xyz
x y z
T
25. Cho cc s thc dng x, y tha mn: 1 1 2x y . Tm gi tr nh nht ca biu thc:
3 3 2 2( 6) ( 6)A x y x y y x
26. Cho 3 s thc dng x, y, z tha mn: x + y + z 1. Tm gi tr ln nht ca biu thc:
P = xy + yz + zx 2xyz
27. Cho cc s thc dng x, y, z. Chng minh: 3 3 3 3 3 3
2 2 2
5 5 53 3 3
x y y z z x x y zxy x yz y zx z
.
28. Cho ba s thc dng a, b, c tha mn: a.b.c = 1. Tm gi tr ln nht ca biu thc:
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
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ab bc caTa b ab b c bc c a ca
29. Cho hai s thc x, y khc khng, tha mn: 4 2x yy x y x . Tm gi tr ln nht v gi tr nh
nht ca biu thc: 2 2 3T x y x y
30. Cho ba s thc dng x, y, z tha mn 3x y z . Tm gi tr ln nht ca biu thc:
2 2 21 1 1T x x y y z z
31. Cho cc s thc khng m zyx ,, tho mn 3222 zyx . Tm gi tr ln nht ca biu
thc
zyxzxyzxyA
5 .
32. Cho x,y R v x, y > 1. Tm gi tr nh nht ca 3 3 2 2
( 1)( 1)x y x y
Px y
33. Cho x, y, z l cc s thc dng ln hn 1 v tho mn iu kin xy + yz + zx 2xyz
Tm gi tr ln nht ca biu thc A = (x - 1)(y - 1)(z - 1).
34. Chng minh: 1 1 1 12x y zx y z
vi mi s thc x , y , z thuc on 1;3 .
35. Cho ba s thc dng a, b, c tha mn 3 3 3
2 2 2 2 2 2 1a b c
a ab b b bc c c ca a
Tm gi tr ln nht ca biu thc S = a + b + c
36. Cho x, y l hai s dng thay i tho iu kin 4(x + y) 5 = 0. Tm gi tr nh nht ca biu
thc S = 4 14
x y
.
37. Chng minh rng: 2
cos 2 , .2
xxe x x x R
38. Cho a, b, c l 3 cnh ca tam gic c chu vi bng 3. Tm gi tr nh nht ca biu thc:
3 3 3( ) ( ) ( )
3 3 3
a b c b c a c a bP
c a b.
39. Cho ba s thc a, b, c tho mn abc= 22 . Chng minh rng:
4224466
2244
66
2244
66
acacac
cbcbcb
bababa
40. Cho a,b,c l ba s thc dng ty tho mn a+b+c = 2 .Tm gi tr ln nht ca biu thc :
2 2 2ab bc caPc ab a bc b ca
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
50
41. Cho x, y, z l cc s dng tha mn x y z xyz . Tm gi tr nh nht ca biu thc:
2 2 2
1 2 5Px y z
42. Tm gi tr ln nht, gi tr nh nht ca hm s
2 4
22
3 8cos x 12cos xy1 2cos x
43. Tm GTLN ca 4 4
4 4
2( )1
x xf xx x
44. Cho cc s thc dng a, b, c. Tm gi tr ln nht ca biu thc
2 2 2
1 21 1 1
Pa b ca b c
45. Cho a,b,c dng tho mn 2 2 2 1a b c . CMR: 3 3 3
2 2 2
1 1 1 2( )3 a b ca b c abc
.
46. Cho x,y,z dng tho mn 3x y z . Tm GTNN ca biu thc
3 3 3
4 4 4
2 1 8 4 2 2 1 8 4 2 2 1 8 4 2
x y zPy y x z z y x x z
.
47. Cho a,b,c dng tho mn 2 2 2 2 2 2 2 2 23a b b c c a a b c . Tm GTLN ca biu thc
2 22009 2011 2007( ) 2009 2011bc a c a b c bc a bPa bc
.
48. Cho x,y,z khng m. Tm GTLN ca 1 1
1 (1 )(1 )(1 )P
x y z x y z
49. Cho , ,x y z l cc s thc dng, tho mn 3x y z . Tm gi tr nh nht ca biu thc 3 3 3
(2 ) (2 ) (2 )x y zP
y z x z x y x y z
.
50. Cho ba s dng a,b,c c tng bng 3 . Chng minh rng
3 3 3 3 3 3
2 2 2 2 2 2
3
3 4 11 3 4 11 3 4 11 5
a b b c c aa ab b b bc c c ca a
51. Cho a, b, c l ba s dng tha mn 12
a b c . Tm gi tr ln nht ca biu thc:
a b b c b c a c a c a bP
a b b c a c b c a c a b a c a b b c
52. Cho x > 0, y > 0 tha mn 2 2 3x y xy x y xy . Tm gi tr nh nht ca biu thc
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Chuyn Bt ng thc LTH nm 2015 Bin son: Thy L Xun i cvp
51
22 2 (1 2 ) 3
2xyP x yxy
.
53. Cho x, y, z 0 tho mn 0x y z . Tm gi tr nh nht ca
3 3 3
316x y zP
x y z
54. Chng minh 2 2 2 1
2a b c ab bc ca a b c
a b b c c a
vi mi s dng ; ;a b c .
55. Cho a,b, c dng v a2+b2+c2=3. Tm gi tr nh nht ca biu thc 3 3 3
2 2 23 3 3a b cP
b c a
56. Cho x, y, z l cc s thc dng ln hn 1 v tho mn iu kin 1 1 1 2x y z
Tm gi tr ln nht ca biu thc A = (x - 1)(y - 1)(z - 1).
57. Cho cba ,, > 0. Chng minh rng:
3333 )111)((3cba
cbaac
cb
ba
.
58. Tm GTLN ca hm s 2 2 22 4 2( ) sin os 2sin
1 4 3(1 4 ) 3(1 4 )x x xf x cx x x
.
59. Cho x, y, z, t, s l cc s thc thay i tha mn 0 x y z t s v 1x y z t s .
Tm gi tr ln nht ca biu thc s ( )( y )zt ts ztP x yy s xz .
60. Cho , , 0x y z v 34
x y z . T m gi tr ln nht ca biu thc:
33 33 3 3E x y y z z x
61. Cho ba s thc , ,a b c thuc khong 0;2 v tho mn: 4ab bc ca abc
Chng minh rng: 2 2 24 4 4 3 3a b c
62. Cho 2 s thc x, y tha mn : 2 2 1 1x y x y .
Tm GTLN, GTNN ca F = 2(1 )( ) ( )2 2
xy x yx yx y y xx y
.
63. Cho 2 s thc x, y tha mn : 2 24 1x y . Tm GTLN, GTNN ca 2( 1) 4 ( 1)
2( 1)x y x yP
x y
.
64. Ty theo tham s m, tm gi tr nh nht ca biu thc P= 2 2( 2 1) (2 3)x y x my . Vi
,x y
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65. Cho 2 s thc khng m x,y tha mn x2 + y2 + xy = 3. Tm gi tr ln nht , nh nht ca biu
thc P = x3 + y3 ( x2 + y2).
66. Cho a, b, c, d l cc s thc khng m, khc nhau tng i mt, tha mn iu kin
ab + bc + ca = 4. Chng minh rng 1)(
1)(
1)(
1222
accbba
.
67. Vi a, b, c l nhng s thc dng, chng minh rng
cbaabbac
caacb
bccba
1
535353 222222.
68. Vi x, y, z l nhng s thc dng tha mn x + y + z = xyz, chng minh rng
49
11
11
12
222
zyx.
69. Cho cc s thc khng m a, b.Chng minh rng:
( a2 + b + 43 ) ( b2 + a +
43 ) ( 2a +
21 ) ( 2b +
21 ).
70. Cho cc s thc dng x,y,z . Chng minh rng: 0222
xzzxz
zyyzy
yxxyx
71. Cho cc s dng x, y, z tha mn xyz + x + y z = 0. Tm gi tr ln nht ca biu thc:
P = 1
21
31
2222
zyx
72. Xt cc tam thc bc hai f(x) = a x2 + bx + c, trong a < b v f(x) 0 vi mi x R.
Hy tm gi tr nh nht ca biu thc M = ab
cba .
73. Cho ba s dng a, b, c tha mn iu kin : ab + bc + ca = 2abc.
Chng minh rng: 222 )12(1
)12(1
)12(1
ccbbaa 21
74. Bit rng 3
21 yxyx . Tm gi tr nh nht ca biu thc M = 1
12
1
yx
.
75. Cc s dng a,b,c tha mn abc = 1.Chng minh rng: )(23111 222 cbacba .
76. Cho ba s khng m a, b, c tha mn: a + b + c = 1. Chng minh rng: 2a + b2 + c2 + abc12 1
77. Cho x, y l cc s dng tha mn 3111 yxxy
. Tm gi tr ln nht ca biu thc:
M = 22111
)1(3
)1(3
yxyxxyx
yxy
.
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