1. Trang chủ
  2. » Tất cả

Ebook những viên kim cương trong bất đẳng thức toán học phần 2

20 3 0

Đang tải... (xem toàn văn)

Tài liệu hạn chế xem trước, để xem đầy đủ mời bạn chọn Tải xuống

THÔNG TIN TÀI LIỆU

Thông tin cơ bản

Tiêu đề Ebook Những Viên Kim Cương Trong Bất Đẳng Thức Toán Học Phần 2
Trường học Đại học Sư phạm Hà Nội
Chuyên ngành Toán học
Thể loại Sách điện tử
Năm xuất bản 2023
Thành phố Hà Nội
Định dạng
Số trang 20
Dung lượng 798,98 KB

Các công cụ chuyển đổi và chỉnh sửa cho tài liệu này

Nội dung

Chlrrrlzg IV 1\‘l117''''ng rién /cim clrrmg Irong I211! (hing I/1121'''' /zifn 11111 5 19 §1s PHU''''()''''NG PIIAP DON BIBN (MIXING VARIABLE) T6111 tét n51 dung 1 11511 1 511 114% II B511d5111g1h11''''c 11 1 111611[.]

Trang 1

ChlrrrlzgIV: 1\‘l117'ng rién/cim clrrmgIrongI211! (hing I/1121' /zifn11111 5-19

1 11511 1-511 114%

II B511d5111g1h11'c 11:1 111611 \0'i @111" 111 11111 1111 gi1111"1 1115111151 \11'11g

1.B1§1d1'111g1l11'1'c 1\'11511g 111151110511

Z D511 bién \/0'1 1151 112'11g 1h1'1'c 00111511 kién

3 D511 111511 111'5'11g gizic 11'011g 111111 g111c

III. D511 111511 b§111g 1\'§111uf1111Z1111

IV B211 111'111g 111110 1111 bién \'0'i c1_1'c 111 11111 d11'5'c 1111 111611.

v. 911111 1y s.\1\1> B5111.111g1111'1~1~ 116111 111é11

VI D5n hién h§11g11Z1111 151

v11.1)611 111é11 11-11111111 1111 111111111f.\~1v>

IX D511 bién bng quy 115111 1111111

X D511 bién101111 1111511 11i.\1'1\'1

1 E.\11V \'0'i 11151113111)

Z. EMV \'0‘i 111511 11'0ng111n1 gizic.

X1 .\11f~1 11-1e11 11611 1111-i~11 111j1¢ 111111.

XII. D11111 1)‘ 11511 111511 15ng 1111111 (G.\IV1

1. DAT VAN DE

c1ic'b111 diing 1111'1'c 1151 x1'mg h11y hon vi 11511 C0 C15/111g 1111'1'c xz1y Ia khi c1ic bién $5 bng nhuu M51 vi d11 kinh 111511 111 11111 115111;: 111110 AM —GM . chng hgn \1'0'i /1 : 3 111 C0 11511 d€111g1h1'1‘c s1111xé1y1"11c1é1ng1h11'c khi \'i1c111'k11i \*=‘\'=: 2(1:

;z3-° S5 1u'5'11g c11c 11211 dng 1111'1'c 111111" 1151' 11hi5u dén 1151 121111 c111111g 111 1u511 1i11 ring C15111g 11111'c xziy 1'11 khi \"£11;111 khi 151113 c1'1c 111511 <5 bf'111g 11111111 111 11151 diéu 111511 11111611. \‘h5n X61

$111 15111 11Z1)' c0 111é 111511;; cam 111101‘ 1101 \1 \*i{.>c x1"1)' 111111; 11151 11:11 11§111g 111111‘ 1151 x1'1'11g

110510 h01111 \‘1 $110 c110 déng 1111'1'c 15111 Q0 111% \a_\' 1'11 151 11311; 111111 151 ca c1'1c hién $5 1\'11511g

11111131 11111111 1111 (101 1101 11g11'0'i 111111 101111 1111111 Q6 11151 1111111 C15 1'51 c1111_\511 11g11i§p C112'111g 11311 d11'é'i

dy 111 11151 vid1_11i€u 11ié11c1101111§111 \é1 1111)":

B511 1.2 [v1\110] c1111 1-__\".;e 21.1-’+_1»1 + -P =9.1<111¢11s 21.1-+_1~+1)-.1-1-;§11>.

T1‘011g b€11d1-“Ing 1111'1'c 1111)-' 1111 c1?111 "1" .111; 1‘11<:> (11 _\" :1 1Z1 1 1101111 \'1 0111112; 21-1)

Trang 2

3)() P/zu'u'ng plzdp WV

' C6 thé nhiéu b2_1n sé ngglc nhién h0‘n khi biél réng cbn en nhfmg héi déng llu'rc mil du

-‘—'——+~!l~+~—‘?§%\1¢1+/v+<*

U dZ1y.(i5L1 déng lhtrc my 111 khi (1: 31> > 0 1' =()(\'Z1c;'1_c dang hozin vi) (‘zic han c(v1]1C' tn‘ héi Ii: czic gizi lri chng hgn nhu‘ (3 1 O) c6 gi dzfac hiét mil liun chm déng lhirc xay ru.

Métczich 111_rc gizic la thiy dumg nhu" diéu dfxc biét G6 121 do c6 mél bién hing 0. Vi gia thiét 1Z1 czic bién $5 a I2 (151 caic thuc khéng fun nén bién hing () ciwn du‘<_>‘c goi IE1 bién c6 giétri Gtrén bién

° ~M©t diéu thL'1 vi nfra 151 czic ban sé cbn gp nhfrng bait dfing lhirc mil du "='" xny ra nhiéu h0'n mét Chg (khéng tfnh czic hozin vi) Chdng déu 151 czic bnl ding lhirc dqp vii

kh('>_ chng hgm nhu'm<f>1 bin lozinrl n6i liéng sau dy: T

6 dy, ding thfrc xciy ra khi a = 12 = 1" > O hozflc (1 = 12 > 0 0 ‘= () (vfx czic hozin vi) Tuy nhién néu tinh czic ban sé nhzjn ru rng du ding thfrc X3)" ra lrong l1'u'(mg hqp nE1y

khéng nm ngoili czic tru'b'ng hqpG" czic vidu u'u"(>‘c.

' Tém lai, trong lhé giéri bit dz“i11g thtrc czic u"n'<\'ng hqp du "=" tln1"6>'11g xay 111 lhco min

+Tru'C>'ng hqp tét céczic bién $5 bng nhau la sc gcpi 121 "cu‘c1ri dat du'Q'c lai lf1n1".

+Tru'b'ng hQ'p khiC6 mi)! biénQ6 gizi U1lrén hiénta sé g(_>i In “cgrctri (Lin du'Q‘c l;_1i hién“ Phu'0'ng phzip dén bién duqc dél ru dé gizii quyél czic hit dng thtrc Q6 dang nhu‘ trén

Bing czich dun vé tru"‘0ng hqp c6 hai bién bng nhuu hozjc Ii: min bi€§n 06 gizi lri mi bién

du Bic biét néu bit ding thfrc mdi chi cim mcfvl bién thi n6 c6 thé du‘<_>'c chtrng minh den gién béngczich khzio Sm ham mm bién

' By giy chdngtéi sé trinh bily céc ky thuét chfnh cua phu'0'ng php Lhéng qua czic bfli toein cg thé Tru‘c'>'c hét déi tnrcyng r-it quun lrqmg mffchnng téi muéwn ban doc n2'\m bit lil

czic bit ding [hirc v('>‘i babién Sau G6 121 czic md réng cho h<"§n bién $5 vZ1cu<">i cimg IE1

phu‘o'ng phzip d§'>n bién tfwng quail cho n bién s5. Trong phn cuéi nixy. chfmg 16i sé gidvi thiéu czic kélqué"c6dién".1'5i dén nhfrng saingI§10nhO vil kél thflc 121 nhfrng kél qua hét

u_1‘ nhién Qua vain dé. Qua d6 czic ban 1)‘ giai du'Q'c "tai @210" dé révi c6 thé tn‘ minh bu'é'c di trén con du'€mg szing 1:10.

1

Trang 3

C‘/zmmgIV:.\'l1'ng vién kilnz'1r0'n_Q {rungI20’! z/(7/lg I/1121'/zifn dqi 55 I

||. BATDANG THU’C BA BIEN vow cue TR;DATTAI GIA TR! BIEN DO! xuwe

° Gia su' In cn chtrng minh h-{at dng lime _/'(.\".’\*.:)2() \"o'i ,\_ _\". : Iil czic hién sf» lhgrc

thou mam czic unh chat nim day Khi G6 la \E‘ lhuc hign mi vL1'O'ccl11'nh mu:

bién sun chm IWQ Q» (I I. i) H1011 min mQi lfnh chincL1z1l\®(.\'._\.:.i

Bmir 2: [);'mhgi;i_f(1 /. t) 2 ().

hing czich chufin hm mic bién l1'on§_' hit dzing lhfrc ll‘L1‘O'c khi llmu hién hui l\Lrb'c.

~

Pt»

G

Iv

Dfvi \=('\'i nhfrng hl ding lhtrc kh<3ng Q6 diéu kién Lhi dim bién Lhco czic dgai lm_)'ng [rung

bmh Chang hqn I=%';I = \,='.\"_\‘:I =\§?;— l;\ k} lhuql chmh dung dc don hm blen

hing nhzlu.

T11 xél m<}l $6 bili main:

Bi 2.1 Cho .\', _\‘. :2 (). Chtrng minh rang: x + _\" + :_ 2 3~ §-“,\‘_\": (L 1)

Ch12'/lgminI1

Bu'0'c 1: Chung mlnh: ‘/ (.\‘.>\'.:)2_/ (/.I.:,) \"0'1 1:7‘. I11 co: 1‘ 21\'_\' suy 111:

_/'(.\‘._\'.:.)—_/'(l.l.:):.\'+_\‘+:—3-{Fill+:-3-§j1:;;:3({'i—§’:F)2()

Bu*6'c 2: Chfmg minh _/'(L1.:_) =21+;— 3~ 20.

'I’hz_‘1t\‘z_“1ylz1c6:_/'(r./‘.:)2()<:>(Z1+:) —Z7l‘:2()<:>(/—:)'(8z+:)2() (dflng)

Criclz 2.- <1) <=> ‘/‘(.\-._\‘.:)I.\*+_\~ +;-3-{A/L\¢T;20 v _\~ _\<. ; > 0.

Bu'é'c l:ChL'1'ngminh: _/'(.\‘ >\‘.:)2/ (/.I.I) \'é'i 1: N/Ty. Tu Q6: Z1 S.\'+ _\- suy 1'11:

Bu'6'c 2: Chung mmh _/(1.1.:)=21+3-3-"\/I"; 20.

Kétluzfm: _/(1-.v\~.:_)2/"(/.1.;)z0

Crich 3: Bu'6'c 1: K)’ lhuz_31l chuén héulhco tfmg d6i \'O‘i bail Cihg lhirc (i5ng hci

ii

Trang 4

553 Phlrrmgphzip J/IV Ti1'd6suy1'u06 Ihé giésu' \"+ V\" + 1:1 (*). khi dé 1,1) Q _/‘ (_\" _\". II — Z7.\‘_\":20

Bu'0‘c 2: y rang khl {hay .\‘ \'L1_\' bow 1:? lhl dleu k1¢n ('*) van du'(_v'c hm) man urc

E9

V0‘1Cl1eu k1¢n("-‘)lh1CIangLhu‘c xay1"uQ 3' Q .\"= \'=% Q .\‘ = \":31%

Vziy lrong tru'O'ng hQ‘p Léng quail ding thirc xix) ru Q.\' = _\" I : > (),

Czich 4: Bu‘6'c 1: Ky lhuétchuén héa Ihco tich d<‘»i \"c'vi bl dfmg lhtrc déng bzic:

TU‘ G6 suy ra c6 lhé gia su .\‘_\":, =1 (*).khid(1(1) Q _/(.\-._\".:)=.\+ _\' +I -320

Bu'é'c 2: Dé rng khi [hay .\" \*Z1‘\' bO'i II\/.\'_\" Ihi diéu kién V5‘) Yin du'<_>'c bao tofm lL'1'c 151

,i

thi \'+_\' 2 2\/.\'_\" I21.11611 \- + >\* + 3 221+ 3 i> f(.\" _\* 3) 2 f(/. I 3).

'»-.

,

Vdi diéu kién(*)tl1idé111g'1hU"c xay 1'11 Q <_ -1‘ Q .\"= _\< =1 Q .\"= _\- I3:1.

2f:

K

Vz_"1y trong tru'€mg hQ'pténg quail ding thtrc my111 Q.\- = _\- = 3 > 0.

a) Qua bén czich Him cua vi du trén ta thy néu sir dung phu"o'ng phzip dém bién '\/Oi kf" lhufat chufin héa thi viéc chtrng minh _/'(.\‘._\".:,)ZO du'Q‘c qui vé chfrng minh hui bl ding

lhfrc \*('>'i111©tbié11 II _f(.\'._\:.:.)2_/‘(1.I.;) : _/'(1.1.:)2O

b) Néu mét bixi tozin d chun héa (u'rc 15 bail démg lhfrc c6 diéu kién) {hi n6 sé "ggvi cho chng ta czich dffwn bién (phzii dam bao diéu kién) I\’gu‘Q"c lzgi mél bi tozin chu"a chuénh6a(b§ld§1nglhl'1'c khéng diéu kién) {hi chdng Lu SE C6 nhiéu czich dé dén bién ho'n. Khi ("I6 ta sé chQn céach C1611 bién suo cho bélo loim (4fU‘Q'IC "nhiéu" biéu lhrc nhzit [rung bit ding thfrc — diéu ny cfmg tu"o'ng during vdi chuin héa sao cho biéu lhtrc c_6 dang don

cn thiél Tuy nhién khi dd qucn vé'i nhfmg diéu n21y tl czic ban <6 lhiyl}-' khmg Q6 su

km biét dzing ké nilo gm Chflng

Trang 5

Chlrr/ng IV:1\’l117‘ng rién kiln c11'(mg[rung111?’! airing tI111'z~ /zifn zI(11' 55?

' Vi hit ding lhU'C AM — (2.11 111 I 31 quzi don gian nén pl1u'<mg phzip d<'»n hién Q13 lhé khéng J6 l;;1i nhiéu n luqng Nhung czic ban >6 hi 1hu)§1 phuc h0"i >11" hiu qua cuu phuvng phzip C1611 bién qua czic \"1'dg1 minh hQ;1 mu df1_\.

(111 + 2)(1>1 +:)(1-I + 2)z@<11/>+1><~+1-11)

Clzzhzg minlz

Do vé mii Qua hél ding lhirc I51 hilm chn \'d'i mic hién 11 /1 1" nén chi cn chfrng minh

bt ding lhfrc cho cdc 56 Lhgrc 11 /1 1' kh(mg m

V ._‘ _ ‘ ,_.

Tiép lhco Lu sé chirng minh _f(1 1 1-) 2 I). Biéu dién f(1_ 1 <') du'O'i dang tam time lwglc 2 {in 1': /'(1.1.<-):(13 +2)“1'3 —1811~+(21‘ —13+8).T;1c(>:

\/§1)'bi1iI0z'1n chL'1'ngminhxong D;'1ngIhL'1'c xa); 111 khi \'Z1ch1 khi11=/2 :1'= 1.

' Nlzgin xét: Viéc gia' SL1‘ 11 = 111111{11 /1 1'} 151 111$! 1hL'1 thujt 151 tl1L1'()‘11g dL1‘<_v'c zip dung dé

dévn bién C511 chL'1 1Z1\'O'i bit déng IhL'1'c hubi§11 déi \L'rng [hi Iu Q6 thé gia sir 11$/251' (hoéc 112l121') cim Lmng 1rL1'(1'11g hgvp hit dng Ll1(1'c bu hién hmin \‘j \"(mg quanh {hi lu

v

Q6 lhé gia $1111 = 111f11{11 /2 1'} 1h0:f1c 11 =11111'.\'{11.

Gidi

Khéng giéunti11l11611g qu;i1_ gié str 11(11+/1+<')2().Xélhiéu:

Dc dimg chung mmh _g>(11)2()bang kl1:10s1i1l1Z1m Lhco bién 11. Do d6 ‘/“((1./>_1')2()

D€inglhL'1‘c xay ru khi vil chi khi 11:/1:010

Trang 6

Plzmmg pluip JIV l§51i2.-1. Cho <1./>.<'Z0 lhcm mfm u/>+./><'+rz/=1 ChU'I]<_Y minh:

_'<1 + />1‘ Z/1 +at 21' — u/7 Kl] +1; + ("J

\/

r

{

Chlingmin/1 D31 _/'(u./5.0):\"T—\/'1’. Tu sé chL'rngminl1: _/'1u./>.<')Z/'(zz././> .1:/%5

Tums hél d5 Ihfi) : 2 D0 \";‘1)' ch: @511 chL'1'ng minh:

Ia‘ +/><‘ L1‘ +1’

-, + , 2 ,- <:>(/>—(') (_/> +_<- +6/n'+u —>u(/>+<~>)2()

Git! sir u=min{u./2.0} :>

Z/>3 + 20: +8/7('+u:—5(1('l>+¢') = I/73 + Z0:+3/70-411: +5(/7~u)(<"—u) 2O

:\ _/‘(mu.lr.(')2() Dz'h1gtl1L'rc xziy 111 <:> 11:/1=1.c* = (H1 2 0) cilng cziu hozin vi

' Nlzgin xét: Bixi toain 2.4 lil m(_3t \'1' du \§ bit ding thirc déi xirng mil cgrc lrj khéng G311 Lzji

Rim. Chfmg In L115)" r§“1ng lrong phu'0'ng phzip dén bién phn quan u‘<_>ng nht 151 dévn vé hai bién hing nhuu cbn suu J6 {hi cgrc lri dql 1:_1i rim hay khéng Kai Liam kh(mg phai I51 diéu mu chét D513 chinh I21 diém khzic biét cun phuvng phdp so \'('>'i viéc zip dung bit dmg Lllirc c6 dién \'Z1 n<'>i chung \"(')'i czic dang Lozin mix cgrc tri khng Gal Lai lfun {hi \u' I)‘ hng bl ding

lhfrc c6dién1Z1lu'o'ng déi kh(> khfm

.

\

~

-.

C

7.“

-.

-fv)

-I-I

2 Dfm bién vé'i bait ding th1i'c cé (1

Khzic vdi plln Lru‘é‘c O‘ phn ni1)' khi X61 nhfmg hail ding lh(1‘c Q6 diéu kién czich Lhtrc

dém bién Qua chflng in sé Q6 diéln khzic. Chng h:;m nhu" \'O‘i diéu kién (//>+/1<'+m:I.

khi chflng 1;1mu<‘m (1611 hui bién bng nhuu tfrc d§'\n 4/'(z/ /1. <‘)2_/M././J [hi bién I o'd[1)' khng phai 151 IDQH dai |L1'Q'l1f_1 {rung binh cua l> 0 mil 121 dai ]u\_mg thou min Zul +1: :1.

Téng quail hon:

D5 chtrng minh bél dilng thL'1'c_/'(u />. (J20 \'é'i czic bién u.b.<‘ thou main diéu kién

thda min g(u I. I)=0. Suu dy la} mél vidgx liéu biéu:

U3

E31

ya

P

D31 VT=f(_u./2.0).Gi2'1 sir :1 =min{(1.110} D0 41+/>+<': l nén (13%

Trang 7

C/lu'0’ng IV: ;\'/Ivzg vin kim <'u'0‘ngIrungI711! ¢!17r1_qI/1111' /zifnzlgzi 555

T11 C6: \/<1+1/v~1'1‘ 2Q11 Dod1'1.11é 11511 11i§111;1c111u§111 c111'1'11g 11111111:

\1/)+1<'—(/1: +\/1(‘+1¢1~/>1: Z \Z1/J+1‘1+(ZzI—/5-1‘): <:>(/2-11313-80121)

Du 1113: \’T2v‘<1 +\1°Z(1—u)+11~3u1’ 2\,’3 <:>1113<1—11>1-1~3<112() (dL'l11g)

B511 df111g 1h1'1'c d11'1_1'c ch1’1'11g 11111111. Ding 111110 \;~1} 1'11 <:> (11./1.1")1 —

»—-—'-.

Lu-~

1J,_.

_,_, _

~ ¢

r—’+

J._.

_,._.

',~

B51126 C111» 11+/2+1"23 T1111 1:11 CZ1Ci1L‘ \‘(11111_1'c /1 \;111 C1101

1113+/\)I/13+/1111"‘ <1-/\)21'1+/<1: 11)

Cilii

Bu'é‘(' l: T1'11'(1'c hél 111 Q6 \'Z1i 11111111 x61 dé 111111 g1:111 111311 11511 101111.

<\\

5:

\JJ

Chc_>n ('=0. 11:/1>111z1.x{ suy 111 /1 >0.

H0'1111C1'11,11éu k > O 1h0z1n1Z1111111111c_>i /1" > /1 cf111g1hQz1n1Z111. vi:

1111+/<'>11>1+1'><<-1+1'1={1(+/<1+1’-1][</f+1>+1'-1I<<~1+1>+1’-1]

21

.\11111'\':§y 1110111 0511 11111 Q1/1 111111 111151 1h0z1111f111 11211 101111.

1¢+1»1</>1+1111-1+11+1'_112‘11+11‘+1'_1)=<1+1'>‘

A

Z ¢

B51)" gib‘ 1:1 ch1'1'11g 11> \'13'i /1' >1 1h1 caic 111é1111 115 suu 151 111'1'\'11g d11"o*11g:

(i) (1)d1'111g \'<'1‘1 11191 11 /1 <'1111_1‘c 1h<1;1111f111 <1+l2+¢'23

(ii) (1)d1'111g 161 111111 11./1.<-kh611g{1n1111611111511 11+/1+1" 23

(iv) ( 1) d1'111g \'O'i (1:17I.110 I 3-2.1‘ \'<'1'i 11191 \'E [(1.1].

T11g‘11véy.c11i1;5:u (i)I\(I'1')I>(1'1'i)I>(11‘) 1Z1hié11 nh1é1111én1z1ch1c511c11(1'ng1111nh chiéu

11g11'1_1'c 1£_1i.

Tuch1'1'ng 11111111 111'):>(iii). G111 $1111 /1 1“ 1Z1c[1c sé 111110 k11<311g [1111 111011 111511

11+/>+(' = 3

\/1111111 (T61 x11'ng.111c(>1h§gir'1 s1'1' 0211.19 D (1 3 u +1’) § Z.

11511/‘(u.1>.<~):<<H+1>11>1+11<1-I+11-<1+11‘.11511.1-=#1111 XE 10,11 11211-=.1-21

“"1

\J

;\

J

J

F

V

v12/<—((I%_,t/)1 —c1l22Z /<—(5lf)r—/7) j_ .

Trang 8

553 l’l111'u‘lI;_'plui/I .\l\

_ , , <Z.'~/1—<‘><Z1+"'>+¢-:' I 1 I

—</w~<‘>:<Z/+/>+<-> m/'>~<'):

<\~¢1+/> +\5¢1+('):(/1+¢")‘(ZI1“ m+/>>“<u+<‘>*

Déndf1§‘u1gi;1\L1'u=min{u./>.z’} L111 <13! \Z1 11+/v+('§(\¢1+/> 7* W1 ‘T <-1‘

.\'Ief1Ikhzic.lI1u\' <1:. J xilo ./M/././1\£1 hién dC>i lax c<'>:

Dang lhuc xay 1'z1<:> u :12= <'It hOL_1C u=0./>1:-:1 uungcue houn \';

\'v3

' Binlz Iun: Néu czic hgn chuulh1_1'c \'L_I'lhZ1I1h lhqo kf thut I151)‘ czic ban Q6 lhé Lluln

1111511 héa béld:'111g lhtrc réi l:_1i (1611 hién nhu" ph;-in I1"L1x‘»'c. Chimg hz;1n nhu'h;'11 déng lhU'L‘

lhuan nhal cuubaz 2.8 1:1: (u/>+/n'+m)E 1 , + 1 \ + l _ 2%

Tu)" nhién. nilu czic ban thu' dén bién thco czic (fzgi lu'Q'ng {rung binh thi czic ban sé thy viéc nily

rt phirc tap <i611g thbi hiéu(huyc nét dqp cuu phép dén bién [1100 diéu kicfrn.

Bili 2.9 C110 u./2.<‘20 111021 min u/> +/><- + <11 + (M//7<'I9

' Chirng minh ring: u + /2 + 1" + 3a/2r‘ 2 6

C11/rugminlz

Dill _/'(u./2.0,)=a+/2+(-+311/>0 Gi;'1sL'1'IlZ1bié11tl1Q;1min Z111 -1‘ +611!’ :9 (05133)

Khidét1/=/'(z/_lJ.(")—/‘((1./.r):(/2+<-— ZI)+3(l(/7('—I:)

\/O'i d'i<5:u kién débi1i.vi§c dun czic biéu {hire (/1+1-— Z1). (/n'—/3) \'§d;1ng A(/>—r): lit khzi phfrc lap Do \'51y.chL1nglu sé ml_\" min czich "khéo" hon

Xét diéu ki{:n: ab+bc+041+ulx" I2:1! +13 +60!: <-:>#2+c" — 21)=I: —/7('

u

Dé)'/r2'1ngnéu :> f<1< \»’/2c 3 V61)’ :> ,

,._._J\i -.

~_.

+

, \

By gid ta girl sir (1:min{u.l2.<"}

Trang 9

(‘lz11‘0'n_; I\': .\‘ln?'ngrién kim urrrng {rung/1:7’! (hingI/Mir lzifn dgli 5*“)

w I

(11; 1-]

.v'1;_1l khuc lhzl) (1: —+ \'uh1cndm Incu: _/ (u./.1}: ~—~~~~ ~~~—~;~ -~~~ + (w Z n

Bél dng ll1L'1'c duqc chng minh

Dnglln'1‘c .\'€'1}'1"u<::> u 1/; :1":1 hozjc uI0./2_ :3 cilng czic hoim vi

3 D611 bién l\r9'11ggizic trong tam gizic:

cht'rng minh kl1(mg khzic hil gi xu \"O'i hl dflng lln'1'c dgli \(‘\ nén 0' d[1§ chx xin néu ru Ina}!

\‘L1i \*1'd1_1 dé minhhQL1L‘l1U \"(rc mqnh C1111phuvng pluip {rung dung ht dfmg um nix}.

(mu

\

Ta sé chfrng minh: _/'(;\.B.C)2_/'(?—’\B A B ¢l).T|1;f1t \'{1) la Q0:

<2)

TL‘r(*) I11c6:6co>(‘—c0s(A—[>’)—l23—l—1>() :>(Z)dL'1ng

Dé ma» ;,»(c>z@.|<é1m_>-,1 \(3'i <1) 2*‘/'(.»\.B.C)Z]2—5

7

P = cos A + cos B + cos C

Gizii

Gié str A=mz1x{A.B_C} thi %2A2%. Din .\":c0s%;.\'e [().l]

P2./‘(-\.): 3

Trang 10

560 Plzmrngphrip _\I\

7Q

w r

Q.a;|

.

'7

Dina lhtrc xaw‘ ru <:> A:l;B = C=5 \'Z1 cdc hozin \'1. Vin‘ min P = l +\—'

Bili 2.12 Cho tam gizic ABC khfmg til Chfrng minh 1'§111g:

Clzzhlg minlz

Khéng mill tinh timg qu-£uu1gi:1su' E2A23. Tu \"i§l bt dim: IhU'C lrén du'(>'i dgms:

-1-rOnOd6_f(A B C):sinAsinB+sinAsinC+sinl3sinC

Vi E>Az§ nén ~++‘—"21<>.\-an"-z1.v1\~g1>- (120

2 _> s1nBs1nC Z

Ch(1§'1'§111g:si11: B+sin: C é Zr-osl I1@11[£1Cl1lC§I1 chu'ngminl1:

B§tdfmgIh£1'c nixy dljlng vi Q2A2% Tir My tacé bit dfmg th(1'c cn chfrng minh

Dfing thirc xciy ra <=> AABC 151 tam gizic déu hozfic [um gizic vuéng cim

thirc lrén urong duvng vdi Bail ding thfrcIran I996:

\/

Ngày đăng: 20/11/2022, 21:33

TÀI LIỆU CÙNG NGƯỜI DÙNG

TÀI LIỆU LIÊN QUAN

🧩 Sản phẩm bạn có thể quan tâm

w