toanvdc5 bdt b1 2 3 4 5 am gm

Trang 1

BAT DANG THUC, GTLN, GINN

L.Bak Sr aie, AM- GM va cor bal ding Jur, lu

+ Đát đểng le AN- GM < Casi - SGK ck đủ, 3 a 213

' hb 24+ [Sing

L2 4x kiêu sp da: ki cxzo

bien Rue a=b= c tai Aikin ax aS chur

3 mal ae Fae pl Draws coy < phi cating mí tare Whi dang >

+ Na

Taw: "eSbtkak) > 2b + — =4 >ị+++z AL Met of kot

b | a+b

+ \Jn eet >9O

a2 > 9 Sets Ahk bobble

2 b at+tb+©

+ Me 'Éœ 2b 1xe:@6-b) > oes ð+b >2ds œ 2(2°+) & Gib’ > #ab > ye

+ se! Muse alc tad; 4+ [a- by + (b2 + + L(e-a) 20

& “ty xẻ “4b ~kc|—~>ÐO

a Hal to > ab rbe toca ®O

® @r bro 2 ®(ab+kc + ca)

BS Wire) > 2(ak+ke+«2

« 2(stE+c3)> @œ+k+ S“

Kila 4 ye b= EX) Co WmND > (yea eer ay)” > a(y LLL t 2⁄0 + Bye)

S&S Ceytye + 2x0)” > im “`

% Mot so bai tập

Đ>a L Cù» 2 s6 Ï& x9 khô âm Thea min 7 3.79 GIN ba bide, (Pare

PL 42455 Pay" + TC +

HD

Tac: Pa(eryy +

x~+u

ay Sự tu) Ni xauz 2 vƒ+v ` »o ney

Pe 3x+dÐ + TH les + (At £G)| + (rry)

Số Am bất đáng TÂM» AM- GN va gia that x+y > % da Ake

Plạ 8 + 20) tần = JO

Vay GTN ova P= 40 Ws n= 3, y= 0

% Chu dorry) he fla he 8! an bag do dom wie ki ary = 3

Bai 2.Cho cre 36! thurs daiwy ot 1 fete + AL sẽ Tan GINN cha bite Hate:

ae w++

A= Satayt— + lui 3Ƒ

OH _ HD

Va 289 Aase dahong bo dacd: ar > Nab , toh > a4 > @b(tt pA

Trang 2

rổ > + @ TÂN < Rất Aa Mia b

bf ch (21 ob LAL | | + xr+S +2 aa [ z#+5 tut+2 > Axrty+h aS a @

-(Š + +2~)+ AP +y) + 3G4ry)

Ax 2\Š.x 222 By + Sts 24+A.51+21- 33

he ai far x.y > O ve xu + 4< 20.Tùo GÌNN ca A< XE 2V—

" >on es n+ > eg - ate

» dự

it Ỏ

Ke 2 = z4 aco: A= 2041+ - |1 — No Ai hoa

Dat = >t? | + +>T?<)† a

+ ¿2124 _ 1 ¡ó1 Bd

Pp day Brak airy JÑm AM- ơM G124 SÁ^z2[Ð Lê củ đi -$Š

Này aTNN dœ N- $2 Vé xe), v4

i

Tacs: 46h = SYtiee _ Yaa) + 3ly ew 5 Sy a hy _ 3348 231448

pip Se

- 43

Này 27NN cứ Ra Đn Veh gad, y= At

Poa 4 Choabc 0 vaabrbe tea = 2 CMR:

ParvVaS+ boss tere ce ei rdvee oS ti ze ale

HD

Sui clung baÌ đán du NM- Ge củ 383 faa:

Pe \ TT) Mang +N2Äcai = VR (ab +be + ca) = 243 => dpam

Dor = Ray Ar œ=b-=c=kL

Đài 5 Co ¬,b,L 2C) s5 m8 C1):

aŠ+b`+c°>a{b+c + bÝ{c+a +ea+b

"Tee Jl bie sa theelte a be) ee -bc)_ 424 bes @®

Oo +

XS Vy sab (thes Bal

“T fe op tic sử dung Bab Asay er ng am orbs 4 2\febd - 2alb+e @

lu vz@ > ais bie > 2a bre ®

+ >

Vor 8 Seuss bi+ ote y 2b Veta ®

c`+ arr » 2c\a+b ©

EE

Ca, ,@, Điều về >> 2(a°r'vc”) y 2a(Erc +2EÍcra + 2eÍa+b

e& arbre > a{bt+c +b(cra +c(arb > ảpua

Dal = nổi aot bi az beet 12

Shy tak dey đúc NHTGM4eec đa{iC - 2.Á281xd ¿41 2!S©) 241 (bxec

ab-aa.b 2 2terb 2s, 14 ti D3 2 a

Ate re _ de%, As

ac=zaa.c < =

Trang 3

~abc ⁄ rete

Suy ra: 2a(E+c £ bat rrO)aba stb storie - bel, bie

> aNbye 4 daird (bre) ŒD

2 AL Lt?

Tuong te ¬ 2 4b t2 +4.) ®

Độc g€SE) ©

Cay ®,@,G@ these > Apaw

Da = ndy an Wa äa =|bl—¿ + 32

_ € Yayb 4

ba 6 Che Roeser >O, ==" CNR

KBE fiz chufSe Toon Quing Bids 20237

YD Neg feast tr dary be’)

Taw: + +3 + 22 - +2*# “+ 22(Kryre) = Agia 2z 1 CAI

= Az Iuu + ty ty = HT t3 “Ma Tổ? ?hp

> 2A) + 24 +22 | 65G) @ rapa 7 > Ge cam

ANT mA >O tad (pe + V (AE) ore A |

AE” Axe)”

TAY + 2E +22 (uy ryt 4 zx)”

13 yer 2X t2 — (oy ye £22) ®

Da = oy a 4 P= i =

Prat 7% Che a,b, c > 0 ome: P= | ab tj bc + | fi †^2} _ >

Nz&zo &+ 2z)(c+ 24) V(c+2b)(a+ 2b)

Sử Tự eat az, ony due AN- GM teed:

Qrr(br2) Va@brebe)(ab+2a0 T @b +2bc+@b+22© ab+bc+e+

a "ở Xsw + 22 + +b ads tbe + cœ ®

P (bt 20)(c +20) “ abx+bc+ea ®

CaO _—_ Ca — |

[€+zL)(a+ 2b) 1 Thebe ba | ©

Chiat die [stat asta Cth itn ale be bt thea

“Day = mH Aa kÌ aabed 30

Trang 4

man 22 be Bai 8 Nebaib< dic diay dhea au ab , be , oo = 2.Tw GINN ca bids ue:

T= arbye4 1S

atbre S“ day ak 422) Rb AM- GN tac:

4 db, be),

2l a 1 a& be _

c a

Lert) > TER << A ca alo Ca

IS)? ya ea

Cầu: đc ve! > bey & > arbre > O<a+b+c< 2

pst œ=a+yb+c© > OcR4S

qt ne Ae yw Ocxned

Thcd: T= 4a( 4 +) - 4x The WPT AM-@M và oc ned địa cá -

Chel) 2]

Vasa T-(x+#)+ A Thee KĐT AM- @M và 6< z 43 đaco: ~ L

1» 2 RA + + = 22 Daa lh 7 n= a,bh ds a-b-c-1

0 9 TT _|2?_ L 7 tele

Vay GIN oe T= 22 }ka<b<e =+L

Poai 9 Che 2.y)2 70 hóa vào ory + #T 4 ay Tan GINN cua bidy 48zm|

= A + at + -3

Lab err Gray (E+ 3)"

Thee KÐi ®M— 0M lọc: 204413 1+ 4 > (+0 Hy +4) +E) > ee + Ay + 22

>2xZ+M+2£ < 6

Se dung BOT AM-GN toca:

qty ` eg * 8.7 =1 Gry 8.8

1 Al

3 =— @ 2

yio * raihi fiz a? 2 GIG — 24 S

Piette! [eae ~ fees Ji lại

Cu để» về, và @ >

Bree 4 yo g 7

ä-

> 15» 2

Não @TN8) cá ® =1 kh x - L, Yates

Pai tO|Chalalb > Oo, lab tele lt ITs GUNN leak Ps | + 1

Cc att bee

HO

Cach 4 Tae Pa arbre + 1

pote fit a, |L|1) {

abe arb+c ab “be 7 ea * a+E+c”

See chang ROT AN- GM tac - t+ Mab > 24 Blob = 12

hig if fered + > 18-Ube @)

1

Trang 5

'w@,@ wš ® > tr x.à—» S2) — (ab +be + ca) ®

Theo BHTAN-GN: _A + 3(a+k+¿) z 6

a+Vxc*#

> 1 œ+b+c*~ >»6- 3(+b+c) ©)

TwWAwOs> x 6Ð - 3+ +) - Rifle + be + œ)

<=> Pr 60- 2(a b+ dab + 2be tea) — 63 (ab tbe red

S Pe sÓ- 2(a+b©tc) ~ 21.%.(ak+bc +) = B1- 21 %(2b+ bc + ca)

Jac: (&+k+”“ Bab tbe HPF a + +~c°~ia —bc — ca

Leb d(b-oL (a 20

> —3(ab the+ ca) > -L(d+b+cÐ = =1 2m sktstcteb eH

>> SI-2I tết co)» SỊ -l2Il =

Vay, GININ cua ‘P= 30 Mh a= bee =4

Cade

Sé ding BOT AN-GN te cd: aborbe+ca Tapeh ate %3laL.k.œ« = 9abc

> > = + 7 R: _ | lst

Pe ia a++c*“ "Tư aie Ul i (at on +

> P= @bid.P => dlatb+d” + @rb+" - 16+ 2 tt) ¿ J+ 2{ak+bc+ =)

par y= athe don bel %_—Ì - @-Ð tô=S c€2) xo > eed

Ta aitec Pei IS 49x 4 = Ad + 2(~+‡+) +9n = 19 tafe +@ED |= 20

\lay GNN ote P = 30 bhi a-b-c=

Cad

Vat yy) 7 O Si dng por oe tée:&rgiS( tt +2)> 21agz4129$ =3

rit tt 8 y t R+y+t © Da =o bin=y-2

“ate * a ae ar alec att ab be ca bobted

> Sat aba \a+ TA xbc kea Perera s@b tbc +a)

Jac: G@+bic) Bab tbe hae II Ted’ —|lale “tees

-A 4 (aby +itb -C) h(a) 20

— >> %(ab+ke+ø -(@+bt+c) =1 @

Ap Aury © và @,lf ® > Ðš_ 3+ 4L - 4121-4O

(@x+kx+‹Ý“ 4

\ ay GINN o& P = 50 Whi a-b-c=4

Bail oxi dhp An về 2) +4 <4 Tim GTILN cua P= tary)

rt

Sĩ dug 1 tak hea GM dacs: yr try 7 Ly -alyns Yr dee Daas S516

®- 2x tị )+* “Pr pte oly A

Jad: art = “liên + 2552, Slang BOT ptt GM va o z 1€ †acs

Weta] od Wola bo Ne do hn Atte 4c 2,0 =8

Xây đN ot 1 = SH Min 5, 9-2

Trang 6

Bai l2.Cho abc eo, a+2b64+3c=1.Tim GTLN, GINN cia bit thie

Pez f†? 1?

< Che Than ta NS 20223»

HD

Te g@ thab > Oc a <1, @<b<« HW! @delel

Aco!

+P< +(Œ- Se\re 2 Oo < 2$ 28 (vw ofc & ).Ð.4 Pe

Atrebt+be = 4a44b+ 4c Pere 28 lhio- 4, be 2, c=0

x Jaw: =§&+6b+6c)(a+k+‹)= 47+ 6b + 6c + 3ab+12ke + Fea

TP: - a°+ 4k*~+ ac”+ 4a + 12bc + &cø

> 3P - Sa“ + gle tABcr + lab + 26be + ca

2 = 2a + 9Lk”7+cYt§okb + 2 bc + 12 ca

>9P- 2 - a?+4okFt1sabrlzkerdcazœ > = 2 Wiab,czo)

Dat =lch Wht a bl= 0,0 = +

Vay GTNN cus P= 2 = lÌ› Ta cat

Bat IS Choa ber thia min ath > 4 Tim GINN cáa bữa thie:

P-= 6fWae+) + Š - 3

< Chusin Toon tt Giong 20237

acd: Qa +b) = @rbd+@ b> @rby + 2(a“tk) xÌa+b\ š a+b

> Pz Elarb>+ AG 1 -3 = 16 + @rb| 4+ 5(arb)-%

Si ang BST AM- GM Na gis gót arb >4 > P > 2he +S.4-3 = 95

GANN cua P= 95 bhi a-b- 2

TT

Poa 44 Cho ajc > 0, axb+c=4 Ching sella ig:

T a(sb‹ + )Đ”~ b(aca +1) ott sab i]

P= eortty + ata ti) | ~ Babes) ? I~

< Chuo Toa Las Cai 2023>

HD

4Ð „- dbet\ ys cael SabtA LPL +7, ,#

Cc

SH dug BET At GM td: emp >

yo

C— ae

24 z > + l6

=" 72 TP St tr? r®t 9

a, R 2 hz

et? > Pờzx+au+#

Talc! Ktyrt = 3b++ paced Ca dart = (+34) + ( + 2b) +#+9ð- €(fa+L+©

Se Dung POT AM-GN >

%+9++ +23 x29 +2Á4-6 - 12 > PZIZ

ĐÀ = nản aa has beca4

Trang 7

Dai IS Cho các 36 ture dong Lips thea wan J1+2>3 +2x+2) +2# =5

L) Tim GIWN o&% P= Xt , Yt , 2tr4

a7 tl 2y +| Az4

ay Qi lận BOTAM-GN fod: 4,

lũ kien 3 @x+0@01+1) z ® @

VW ga Wak ta ch: Äzx+0(2u + + “hi

Tra w( 4 A be = Ae 1) + 22 +Ð] > 4

Vex Daytt ae th

A 3 > x = 34 Ê at +

Dar = voy rchhi b0 tÒ =2

z =4

by Tac Po AH, Yt, ere

2P 2a ht ee 2x +1 31! 2z+1 2z +\ A + Ae Ar 1% 4+ z+l +4

> 2P- SS 6 + mm) “li SF dag BAT AM GM Vay ) ta + ứ = a cô:

2P >3 On + D(2y +1 TT ?F 5 Ee Pe

A 9 _ 1» li _ 7 i

Mẫu 0N của P= Mix <a=t z1

Pai 16 Che a,b,c Eat soto Wt al Dan 1 CNR:

vob 11 |_ , Mec=1 ea a+xb+c

< Sr Toan Can The 202 3>

Si dig BE 1 AM- IM sa os: {đaE-\ „ Vab _ A — { =-4- A a=-L)

bre | ae 2 be J <( b

A fA aW Aa:

>, fab-l„ a 4 ft

Vuong WR tac Vbc-l\ „ A (A A

cas} fo Af a tae

Cong thee se O,O.8 > dpom Dar = olia-b-c= (2

Pai It Cho asc >, ab +b} +! cd =1 Tm GILN ca bith thre:

H- 2a + b + —&

— Ä4x~aF— lIx+b” JA+c"

a+ ab tbe +o = (at+byatc’ } 41+k~= V+ak+b©c+œ =(a+by& +c)

7= ch+ab tbc +ca = +cSCb +d

“ladÄv¿tc

Số dung BAT AM— GM taco:

DH 2) 2B 2a xb + ab Ì a+C ae” 2th b

(

NO Ry >O

Yo S| _ are a u(b+

Trang 8

re

> 2H < 2+2++- 2 Snes

as+bc+ca=1 bà TT

arb TL TP c ˆ bec fe Lda

art 2b+e)

we OD tt! biel | i

\Jay GTLN one i na = b c TC

Pai 18.Cho ad, Pp: pete GIUN wa Lier thr:

tr Tat "orb * fare

HD

ee Atay =a + alt +bex+ca=( x+bESCa + c3, 1+E =Q+c)(ax+k)› Aten (bi d(c+a)

“Ja duce

Stag S wa leis an on hes

3b

“r† ate a+b ie 7 it Pate |= TrÊ† 3 &

>t 4 + (Dw = x án ha LÀ Boa = ^^ _ 3a

arb te) > | 3

—= = L = C= Ao

ab | betea = t

œ H- AD -| At ~ LAL

Nay GTLN om R= Oth a = Fe; 2c = te

Bai 19 Cho a,b,c Pur “thea wan (a+) (b Her) = (a- \(b-1)(e-1)

Tim GTNN co A - lal + \bl + Je}

< fe Tin Ha NG 2023)

Tat gid Shick O+DCb+Dlgrd =(a-Nb-N(e-1) & abtbercas-1 O Tad: A = ee alglele te) = = a+b 4+ co + | ab| + bc} + Ica!)

arab tet Lobrbe+ a) +2 + 2(leb\ + Jbc\ + Ical)

@rb+ + 2+2\obtberaal > OF24+21-=4 SASLZ (wWAZOD

le hay an asl, b=t,c=0

Ny GTNIN ofa A= 2 dda chaty than a = 4,bkljcco

Bai 20 Oh a ble >O, ax 2b + ac - 2 CMR

_ „| 2ab +HẴˆh—-+hÀ—-k>~

= VW22E: + 2bc+a ac+?Ùb 2

HD

Dat xnaard,y=d >0, E-48ce>0 SxtyteE=3

TF My + 3% fF yer Sx * EZRHS

Teta dc: 20 + HT CP oe Y=(U4 2) (+E + aie — FY) (x+2), 2% r3y = (x+ylys z BX=@ >)

St dp ob Fore min i Ary Att q lụa _ x+y

P<« th wR ip =o + 4+ ey ) _ 34

As

® io +t Rt t rc %+# b 3” Wie

Sul x a A Ray = + — SM œ= 4; = ;C=—

E

Trang 9

a (2+3)— "Grae (E41) ~

HD

Sit dug BET AN-GM ford: 6u FIA eo AOI + (yb) + ACE HD > BX +BY + RE

> 12 >z4x+u++2z @

g = + K+S + R+3 > => Zz @

Hi t ae t T1 3 5 @

dept etre tial

Nội đTNN cá M= 2024 bl a= 4, y- 4, 2-4

2 Kế thuak Cast ngide dau <AM— GM rgute dad >

La qu Asan; eet AN-GM ae Aduale ki của ae

co

“ A+B

arenes

T= 9 1 „2 Ị

Dba A Chale | 3 CÀI Tim GINN cis bak fue

2= ẩm tự Tự

by Q= k2) + Zz wR

R44 yrad £?+4

D Taw: A =_)+- XL Jaw xt z 2x => aE | Ze lh | oe

wy Ad te tL

J ums Mì 4 _

1 > _| &

i |? 4 1 ®

Céng thes ve! OOO > Ps 3-+(x+Jt2= `

9 _ 8 J _ _ |

Vay GTN ona P = 3 Ue w= ys ea

b> Jac: BY) ye 8 XY Ly 2

~ 9 _ %4)

T wh || 9 | pm]

Tug Ns 2 yg ye yr ~ 2

1L Z z2

® xŒœ-+w+>)- + (xu tuz+ zr) = 3~— 4 (ey ryt + te)

Lại cai (REYt tỳ 3Jr#+z2) = +@œ-vŸ + +- y+ 4@-2)"> eo

> ~+ ry ryt +t) >¬+@x9 xx+) cổ

> AQ» dial 2 =

Vay GTN ota @ = 3 Ws we yee ad

Trang 10

Pai 2 Cho cas $6! Hue aw abe hse ma arbre - 3 CMR

Qe

a+ ob tr + c+ Ja 2 4

chs

Thea _ a(a+ 2b 2b) _ Pate 2ab~ _ ab”

Se ĐỜI AM- ti

OF ¬ > |_| 2 Yar

[ting AS tf: bt2cr 7 bY b-k

c~ z C— + chan

c+2a*“ 5

> pyas- Pe arbre 2 (Nai We + Neat) = 3- = Clot Bot + Wcta®)

TA EIEìN Aa 4 Wer + “jcah 4 3

ar

hott zZ atktab

“| 3

® ber zt bactbhe

3&

= 2 crater

Cs: @+b+‹3 ~3(ab+bc+eO = 1@-)T+ ?ứ-ở++(-sŠ > O

>> eb +bc +a) 4 %(& +tb+9ˆ= +

TO EO > > HAS aig ails

Gok sb Gs

@®

As

Bas Che abc >O Rea TT + Ạ lực HT TT TU ‡ + z1 tng mink Aan l:

a+2b° b+2<5 c+20° ~

Tiêu đề	AM-GM Inequality
Tác giả	L. Bak
Chuyên ngành	Mathematics
Thể loại	Notes

Định dạng
Số trang	10
Dung lượng	14,11 MB