(2x + l + 2y + l)^
(Do (2x + l)(2y + l)<-!^ _ ^ = (x + y + l ) ) Mat khac ta cung c6: x + y < v/2(x^ + s u y ra
P > 1 + -
\5777)+l 2( x 2 + y 2- l ) Dat t - /2(x^ + y^) . T u gia thie't ta c6:
( x 2 + y 2 f+ 4 : . 2 x 2 y 2+A = 2 x V + — + — + — > 2 + 6 = 8
\ I xy xy xy xy
2 2 2
xy xy xy > 6 . N h u vay x ^ + y ^ > 2
Khao sat ham so f(t) = 1 + — — vol t > 2 ta t i m duoc
t + 1 t^_2
1
m i n P = — khi x = y = 1
3
17) A p d u n g bat dSng thuc Cosi ta c6: ^- ' i •
a + b + 4c â + b^ + 4ac + 4bc
(a + b)7(a + 2c)(b + 2c) <(a + b)- <2(a2 + b2+c2) Dat t = j f a ^ +b2 +cM + 4 > 2
Suy ra f(t)<- 2 = f(t)
t 2( t 2- 4 )
Khao sat ham so f(t) tren ( 2 ; + o o ) ta c6
.3 , ^,2 t^ ( t 2-4) 2 t^ ( t 2-4) 2
Do 4 t- V 7 t 2- 4 t - 1 6 = 4 ( t ^ - 4 ) + t ( 7 t - 4 ) > 0 V t > 2
• ị i t , . ' I
- / (1)1 M Y I , . '
Lap bang bien thien cua ham so f(t) ta suy ra P < f(t) < - . Dáu bang xay ra
8
khi a = b = c = 2
8)Tac6 . Q - P - l ^ - C ^ ^ ^ - ^ ) , 12(a + c - b ) ^ 25(a + b - c )
Dat b + c - a = 2x a + c - b = 2 y : a + b - c = 2 z a = y + z b = X + z c = X + y T , , J... . . . 4b 3, 4b T u d i e u kien: — > a - c > - b = > a < — + c < b + c = > 0 < 4 x < z < 9 x 5 5 5 24x 24z 50x
Ta CO 49 - P = A = + + . Ta coi day la ham so cua v v o i
x + y z + y z + x ^
f, , 24x 24z 50x ^ 24(z - x)(y2 - zx)
%) = + + y r ^ ; f ' ( y ) = O o y = ±V^
x + y z + y z + x (x + y)2(y + z)2 ' ^
De thay y -> ± o o thi f(y) > 64 . Ta xet tai hai gia trj y = ±>/zx
Tfnh f(-V;;i) = ^ ^ + ^ ; f ( V ; ; ^ ) = 4 8 ^ ^ ^ ^ ^ i m ^ _ 5 0 _ V x - V z z + x Vx+^/z z + x t + 1 t ^ + i
1 1
Khao sat ham so f(t) suy ra f(t) > f v2y ' I ' - 56 => P < - 7 .
Chii y: Ta chi can khao sat ham so f(4xz) v i de thay f(Vxz) < f ( - V x z ) .
19) Ta CO â + + c^ = (a + b + c)^ - 3(a + b)(b + c)(c + a) < 27 - 24abc Suy ra ta c6: Suy ra ta c6:
â + b^+c^ ^IzZff^ + 8 ^ = ^ ^ ^ ^ + 8 ^
Toi day chi con moi mgt bien abc nen ta nghi tai phuong phap ham so de khao sat ham mpt bien. khao sat ham mpt bien.
Dat: t = abc .
Suyra f(t) = ^ / 9 ^ + 8 ^ voi t = abc < ( ^ ^ ! ^ ^ ) ^ = 1 =^ 0 < t < 1 '
Ta c6: f (t) = -8 8 8
^ ^ ( 9- 8 t r
3^(9-8t)2 3 ^ 31
Vi (9 - 8tf - 1 ^ = (9 - 9t)(9 - 7t) = 9(1 -1)(9 - 7t) > 0 =^ f ( t ) > 0 Suyra: f ( t ) < f ( l ) = 9 Suyra: f ( t ) < f ( l ) = 9
Suy ra ta c6 dieu phai chung minh. Dau = xay ra khi a = b = c = 1 Dau = xay ra khi a = b = c = 1
20) Gia thiet viet iai thanh : ^^(^V + +^'^) = 5
(x + y + z)
r . - ! 4x , 4y 4z . , ^ Dat: a = , b = , c = thi ta co: Dat: a = , b = , c = thi ta co:
x + y + z x + y + z x + y + z a + b + c = 4, ab + be + ca = 5 a + b + c = 4, ab + be + ca = 5
Ta phai tirn BGTLN, G T N N ciia bieu thuc: P = 4 1 1 1 P = 4 1 1 1 a b c j 20 ăâ - 4 a + 5) • = f(a) 2 Ta de thay : (b + c)^ > 4bc => {4-af > ^á^ - 4 a + 5 ) = > - < a < 2 Den day khao sat ham so f(a) ta tim dugc GTLn, G T N N <
4z 4x 4y 4x 4y Chii y: Vifc d|it a=-—: , b = -,c =
x + y + z x + y + z x + y + z thuat chuan hoa de chung minh bat dang thuc thuat chuan hoa de chung minh bat dang thuc
thuc chat la ky l A / I l A / I 21) A p dung BDT A M - G M thi ta c6: Do do ta c6: P > • 1 1 - + (1 + b ) ' ( l + c f (b + l)(c + l) (l + b)(l + c ) < l ( 2 + b + c)2 16 (1 + a r (2 + b + c)2 (l + a)(2 + b + c)2
Mat khac tir gia thiet ta c6: b + c = ăb^ + c^) > - a ( b + c)^ => b + c < -
2 a D o d o t a c o : P > — L _ + § — _ + 16 2a^ + l 43^ D o d o t a c o : P > — L _ + § — _ + 16 2a^ + l 43^ .— ^ + - 2â +6â + a + l (a+ 1)3 5a+ 1 = 2 - (1 + a) 5a+ 1 (a+ 1)3 2 + ± a
Xet ham so: f(a) = 2 Va > 0 => f(a) > f (a + l)> (a + l)>
Dau "=" xay ra khi va chi khi:
f-1 .5] .5] 91 108 (a + 1)^ (a+ 1)3 1 a = — 5 b = c = 5 , J , . Kétluan: MinP = _91_ 108 2) Ta CO Vsbc = 2'Jh2c < b + 2c Suy ra Mat khac, 72(a + c)2 +2b^ > (a + c) + b -8 . -8 Suy ra 1 a = — 5 . b = c = 5 • 1 2a + b + x/sbc 2(a + b + c)' 3 + ^ a + c)2+2b2 3 + a + b + c •P> 1 8 2(a + b + c) 3 + a + b + c D$t a + b + c = t, t > 0 . Xethams6 f(t) = -^ t > 0 . 2t 3 + t Taco f ( t ) = - i - + — ^ = ^^^11(^11^, t > 0 Suyra f ( t ) > 0 « t > l . 2t2 (3 + t)2 2t2(3 + t)2
N h i r v a y P>- — .
Dau dang thuc xay ra k h i
23) Gia su 0 < a < b < c < 3 Suy ra a ( a - b ) < 0 a ( a - c ) < 0 a + b + c = l b = 2c b = a + c a ^ - a b + b^ < b ^ ^ - a c + c^ <c^ ( b - c ) ^ - 3 b c a = c = — 4 <=> i D o d o P< b 2 c 2( b ^ - b c + cO = b^c T u 2„2 ^ ^ " ^ ^ ^ ^ t a c o b + c < a + b + c = > b + c < 3 < : > 2 V b ^ < b + c < 3 0 < a < b < c < 3 9 Suy ra 0 < be < - T u do t a c o P< b 2 c 2( 9 - 3 b c ) X e t h a m s o f(t) = - 3 t 3 - 9 t 2 v o i 0 < t < ^ =^ f^(t) =- 9 t 2 + 18t Lap bang bien thien cua h a m so f(t) ta suy ra P < f(2) = 12 Vay G T L N P = 12 k h i a = 0; b = 1; c = 2 va cac hoan v i
a2 ,,2 j,2
24) Theo bat dSng thiic Co si ta c6: — + b > 2a;-— + c > 2 c ; — + a > 2c t u do ta 2 2 2
suy ra — + — + — > a + b + c .CGng theo bat dang thuc Co si ta c6: b c a (2a + 2b + 2 c f 8 . ^ x3 , ^^{^ + ^ + ^f + c)(c + a)<-^ (a + b + c) v a a b c < ^ - ^ ^ . (a + b)(b T u do ta c6: A > 27 2 7 ' 9 (a + b + c f + — ( a + b + c ) - + a + b + c. 27 Dat t = a + b + c suy ra t € ( 0 ; 3 ] . K h i do ta c6: A > t + — = f(t) Xet h a m so f(t) = + t f e n t € (O; 3' t ^ ' - S l
Ta CO f (t) = — < OVt e (O; s ] nen h a m so f(t) nghich bien.
t S u y r a f ( t ) > f ( 3 ) = 4 , Vay m i n A = 4 k h i a = b = c = 1 4 a b — + — b a 25) T u gia thiet ta c6: 2
Theo bat dSng thuc Co si ta c6: (a + b) + 2 l = (a + b) + 2 —+ — 1 ^ a b ^ 1 1 ^ —+ — a b > 2 2(a + b) 1 I a ^ b = 2. 2 b ^ a ^ a ^ b a b + 1 > 2 J 2 f b ^ a ^ fh a] — H + 1 > 2 J 2 + 4 <=> V ^ a ^ b ; . a ^ ' b . Tir do suy ra 2 Dat t = ^ + - > | t h i A = 4 t ^ - 9 t ^ - 1 2 t + 1 8 - f ( t ) . + 4 . > 5 2 ' 5 ' v 2 y 23 Khao sat h a m so f ( t ) f(t) > f
26) T u gia thiet ta suy ra 0 < a < - 5
Ta viet lai A = ăb + d)(c + e) + cd(b + e - a) Theo bat dJing thuc Co si ta c6:
{b + d + c + ef ( i _ a) 2 (b + d)(c + e ) < - ^ (c + d + b + e - a f ( l - 2 a f cd(b + e - a) < -^^ = ^ L- ' ' 27 27 S u y r a A < — ( - S a ' ' - 6 a ^ + 3a + 4 ^ 1 0 8 l /
Khao sat h a m so f(a) = -5â - 6â + 3a + 4 tren 0 < a < - ta c6 5 f ( a ) < f . 5 , 108 ^ ' 1 => A < — 25 25
Dau bang xay ra k h i va chi k h i a = b = c = d = e = — 5
27) Ta thay vetrai la bieu thuc doi xung 3 bien
Ta c6: + b^ + - 3abc = (a + b + c)(â + b^ + - ab - be - ca) (a + b + c)(â + b^ + - ab - be - ca) (a + b + c)(â + b^ + - ab - be - ca) Suy ra: VT = (a + b + c) 2 - (a + b + c f - 2 Dat t = |a + b + c=>0<t<\/6 Ta C O VT = t 2 - t^-2 : - - + 3 t 2 Xet ham so f(t) = - y + 3t,t € {0;S] ^ f (0 = -|t^ + 3,f'(t) = 0 o t = 72
Lap bang bien thien ham só f(t) ta suy ra f(t) <i^\l2^ = l-Jl
28)Kh6ng mat tinh tong quat, ta gia str a > b > c
Dat f(a, b, c) = ab + be + ca -12 (â + b^ + )(âb^ + b^c^ + c^â) Ta se chung minh: f (a,b,c) > f (a,b + c,0). Ta se chung minh: f (a,b,c) > f (a,b + c,0).
Thatvaytaco: f(a,b + c,0) = ăb + c)-12 â+(b + c)^ â(b + c)^ Mat khac ta c6: ab + be + ca > ăb + c) va Mat khac ta c6: ab + be + ca > ăb + c) va
â + b^ + ê < â + (b + c)^a2(b + e)^ > âb^ + b^c^ + c^â nen ta suy ra f(a,b,e)-f(a,b + c,0)>0 nen ta suy ra f(a,b,e)-f(a,b + c,0)>0
Cuol cung ta chung minh: f (a,b + c,0) > 0 <=> 1 > 12ab(â + b^) vol a + b = 1 (a + b f 1 (a + b f 1
Dat t = ab=^0<t<^^ =
4 4
Bat dang thuc can chung minh tra thanh:
12t(l - 3t) < 1 36t2 - 12t +1 > 0 (6t -1)^ > 0 .
Bat d5ng thue nay hien nhien diing.
Dau bang xay ra khi va chi khi a + b = l ^ r r-
Suy ra dau bSng trong bai toan xay ra khi va chi khi (a,b,e) la hoan vj cua bp so 3-N/3 3-73 bp so 3-N/3 3-73
•;0
29). Khong mat tinh tong quat ta gia su: x > y > z 5 5 Khi do — + _ ^ + _ ^ + ,
x "
i y y - z x - z ^xy + yz + zx
Mat khac ta cung c6 bat dang thiic sau: - + - > ^ > nen ta c6:
A = 2 Hay Hay A>10 1 1 - + • ^ x - y y - z z V^y + yz + zx x - z x - z ^xy + yz + zx
^x-z 2Vxy + yz + z x j ^(x-z)^+4(xy + yz + xz) 7('<+z)(x + z + 4y)
20V2
2V2 20N/2
•V(l-y)(l + 3y)
Theo bat dJing thuc Co si thi
2^3 7(1 - y)(l + 3y) = ^ - 2 7 ( 3 " 3y)(l + 3y) < ^ ( 3 - 3y +1 + 3y) = 7(1 - y)(l + 3y) = ^ - 2 7 ( 3 " 3y)(l + 3y) < ^ ( 3 - 3y +1 + 3y) =
Tu do suy ra A > 10\/6
Dau bang xay ra khi va chi khi
' ( 3 - 3 y ) = (l + 3y) r - r-
\' 2 + 76 1 2 - 7 6 . , , . , , x + y + z = l <»x = — - — ; y = - ; z = — va cac hoan vi cua no x + y + z = l <»x = — - — ; y = - ; z = — va cac hoan vi cua no
6 3 6 x - y - y - z x - y - y - z
30) Dat a = x;b = 2y;c = 3z.
Tu dieu kỉn suy ra x, y, z > 0 va xy + yz + zx = 1 Khi do P = Khi do P =
x^+l y 2 + l z^+l Dey rang xy,yz,zx <1
D5ng thuc xy + yz + zx = 1 giup ta nghi den h| thuc luong trong tam giac: Dat X = tanA;y = tan|;z = t a n | . Khi d6 ta c6: Dat X = tanA;y = tan|;z = t a n | . Khi d6 ta c6:
7 A 2 B 2 C P = cos —+ cos^- + cos - P = cos —+ cos^- + cos -
2 2 2 ^ = 2 - sin^ A + sin A . c o s - ^ < 2 - sin^ y + s i n - = 2 - sin^ A + sin A . c o s - ^ < 2 - sin^ y + s i n -
9 4 4
1 . A
— sin —
2 2 j
Dau bang xay ra khi va chi khi
MUC LUC M
• •
Phan 1: Phuong phap giai phuong trinh, bat phuong trinh v6 ty 3
Ị Nhu-ng kien thuc bo trg cho giai phuong trinh v6 ty 3
IỊ Mot so dang phuong trinh v6 ty thuong gap 6 Phan 2: Phuong phap giai h$ phuong trinh I l l Phan 2: Phuong phap giai h$ phuong trinh I l l
Ị doi xung loai 1 I l l IỊ He doi xung loai 2 115 IỊ He doi xung loai 2 115 Phan 3: Phuong phap ham so trong cac bai toan chua tham so 202
Ị Phuong trinh CO tham so 202 IỊ Bat phuong trinh c6 chua tham só 202 IỊ Bat phuong trinh c6 chua tham só 202
Phan 4: Phuong phap ham so trong chung minh bat ding thuc
CONG TY TNHH MOT THANH VIEN DVVH KHANG VIET <
Nha Sach
KHANG VIET
" 9
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