X goila idean nguyen t6 ngu• va chi lieu veil u, u E tich
vl A[x 1, x2] la mOt vanh non ta c6 phep nhan phan ph6i d6i veil 014 Ong, do do f(x i , x2) con co thg vigt
(3) Art , x2) = c 1411412 +c2421 422 + + einxion 402 vOi cac c i E A, the a n , a i2 la flitting s6 tut nhien va (a, 1 , a,2 )
(ail , ai2 ) khi i x j. Cac ei goi lit the ha to va the cief ief goi la the hqng to cda da this f(x1 ,. x 2). Chang han x6t. da thde
As p x2) E Z[X , x2] vbi Z lit vanh cac s6 nguyan
f(x 1 , x2) = x3i — 1 + + 2).x2 + (4 +a, - 1)x22 = = x3i — 1 + x1x2 + 212 + :434 + ICA —
Da this f(x 1 x2 ) = 0 khi va chi khi the ha t8 c i dm no bang 0 St ca. That Sy nelt the e i = 0 thi 1.6 rang f(x1 , x 2) = O. Deo lai gia f(x 1, x2 ) = 0. Via f(x 1 , x 2) dual dang (1), ta dugc cac da this (2) bang 0 tat ca, We la
b io = = bin = ° =
Dining cac c 1 ,. , c m trong (3) chinh la the
= 7I, do chi
c i = C 2 = ••• = cm = 0.
Bang guy nap ta chang minh mOi da BSc f(x 1, x2 , ..., xn) cda vanh x2 , ..., :r t.] co th6 via dual clang
(4) f(x i , x2,..., x n) = c ixai n ++ cm .491 .../enno vbi cac c i E A, can ai1 , am, i = m, la. nhang so t9 nhian va (a ir ..., am) x (n p ..., khi i x j. Cac c1 pi la the ha td,
cat gra la cac hang td curt da thtie f(x 1 , x 2,..., ES than f(x 1 , x2 ,..., xn) = 0 khi S. chi khi cac ha ti dm no bang 0 tat ca.
Cho hai da thtic f(x 1 ,..., x n) va ex p . x n) bao gib ta sung co th6 via chi-mg olden clang sau day
AS"—, xn) = Xfi:tin + + cnixaimi _49. (6)
g(x i xn ) = + + xna.
Chang han, vol fix,, x 2) =+x ix2 va ,g(x i , x2) =
= ax 1 2 2 x — x1+2 + x2' ta viat
NI, x2 ) =xi + 0xi4 +x ix2 +0x2
g(x i , x2 ) = 0+21 + 2XIX22 — XiX2 +x2
ta c6 tang, hien, tich mitt fix, xn) va xn) la
Do cac tinh chat mitt cac phep than trong vanh Afx„, , ni
fix„„ xn) g(xi, xn) = daxain ...x0„m
txxi , xn) ext , xn) = E cedie,ii+ap +a,,
i,j
i = M ; m
Tit mat da Doi° bang 0 khi vb. chi khi the 110 ti tha ttd bang 0, ta suy ra hai da thdc xn) xn) Wing nhau khi va chi khi thong c6 the hang to y nhu nhau. That vay set hai da thtic fix„, xn), xn) thy y (5). Hien coa chang la
. xn) - xn) = E (c1 —
i=
Do d6 xn) - ex = 0 khi va chi kill c - di = 0,
i =1 tire la f(x,,..., x n) = g(x i ,..., x n) khi va chi
khi c i i = M.
2. BO
Dinh nghia 2. Gia sit /Try x,) E A [xi, xn] la mot da thole klitte 0
f(x„, , = ..yanm + + c,n.X1.01 .••40,1
vol the c # 0, i = 1, ..., M vb. (a u , (a p , khi i # j. of la bdc caa da thdc trap xn) dot veri an xi s6 mu cao nhat ma x i c6 dome trong e hang tit cim da thug. th
Ngu trong da thde xn) an xi khang cd mat thi bac ciia
xn) dal yea nd la 0.
Ta g9i la bac cars hang tee ...4n tong cac s6 mG
aii + + (ti eda cac dn.
Bat cda da Mac (dal bed than thg the an) lit s6 Ion nhgt trong the bac elm the hang tit dm nd.
Da fink 0 la da tilde khang co bac.
Ngu the hang td eita fix 1 , x n) cd ding bac k thi f(x p ..., xn)
g9i la mat da three thing cap bite k hay mat eking bt).e k. Dac biOt
mat clang bac nhat gia la clang tuyin tinh, mat clang bac hai gel la dang than phttong, mat dung bac ba gel la clang lap phuong.
Vi du. Da thdc
Az, , x2 , x3) = 2x 1244 — 4x2 + 34 +4 -5x344-44x3 +6 s b6+ IA 9, nhung &Ai veil x 1 nd cd bac la 2.
DA sap xgp cac hang to elm mat da tilde f(x p xn) khae 0 ta cd thg sap x0p nd theo the lay thita tang hay giam dal vai mat tin flan dd. Chitng han, trong vi du tren ta co thg sap xgp Pa p x2 , x3 ) thee cac lay thita Iui mitt 4.
f(x 1 , x2 , x3) = 4(1 — x2) + x i(2443 — 5x324) + 34 41-2x3 + 5 6 hay theo the lay thila lid dm x2
f(x t , x2 , x3) = — 44x3 + x3(2.x ix53 — —4x2 + + 34 + 6
hay theo the lay thita lui dm x 3
f(4, x2 , x3) = 34 +2% 14%35 — Axi4x32 — 445 —x2ix2 +x2i + 6. •Ngoai cac each sap x6p dd, nguai ta can cd mat each sap xgp gal la each sap xep theo !di tit dien (giting each sap xgp the chit trong td then). Cad) sap xgp nay data tren quan he Qui to toan phan da xac dinh trong tich dd bac (n a 1) yea N la tap hap cae s6 to nhien (ch 1, §2, bai tap 10). Ta co, theo quan he that to cig,
nee va chi ne'u cd met chi se i = 1, 2,..., rt sao cho
a l = ai = 6 1 _ 1 ,ai > bi
Quay lai da thtic f(x i , x2, 1 3) trong vi du tren, the so` mu trong mai hang tit cho ta mOt phan tit thuec N 3 , cu the ta cci 7 phan t8 thuoc N 3 : (1, 3, 5), (2, 1, 0), (0, 0, 9), (2, 0, 0), (1, 3, 2), (0, 5, 1), (0, 0, 0) ma theo quan hO thti ta dang xet - cheng stip thu tut nhu sau
(2, 1,0) > (2, 0, 0) > (1, 3, 5) > (1, 3, 2) > (0, 5, 1) > (0, 0, 9) > (0, 0, 0)
vol (2, 1, 0) la phan tit Ion nhtt. Viet cac hang to throng ang caa f(x l , x2, 13 ) theo thd tren
fix t x2 x3) = xix2 + 4 +20‘144- 6x1x32t3 4r52x3 + 34 + 6 ggi la sap xtp
f(x i , 12 , x3 ) theo lei ta dign.Hang tit -4x2
tudng ling vei phan to len 'that (2, 1, 0) gat la hang td cao
nhdt coa. f(x i , 1 2, x 3).
Dinh li 1. Gid sit f(xi , mOt da the& udi hang to cao nit& g(x i , xn) Id mOt da thtic vat hang tit cao nhdt la dxbi ...xbn. a& gid sU > (bi,..., b,,). Thl thi hang tri cao nhdt diet da the& tang f(xi , xn) + g(xi xn) la all _4,
Chiang mink. Ta hay vitt f(x l , va x) duel dang xn ) = c liqu + + crnxIsi
= d1a1;11 „, xan = dmx7ml xnamn
voi c ix`it it = = 0
i,•••, a,_ 1, > (all , aid i = In .
Ta xn) + xn) =
(c 1 + d i).x7n + + (cm "+ d
Do do hang tit cao nhgt aim da thdc tang lit (e l + 4.ta = ann. n
Ha qua. Old sdft (x1,xn), fk(xl , la nhang da them to hang hi cao ?theft theo the tzt la
celki oh gid sit
(an , ..., am) > (a21, a2n) > aim)
The thi c ixin la hang tit cao nhdt cart da thee tong xn) + + fk(xl, xd.
Tap hop sap the Di IV con la mot vi nhdm giao hoan din via phdp Ong (ch II, §1, hal tap 5)
(a l , an) + (b p ..., b,) = (a 1 + b i , ., an + b.)
Phgp Ong nay va quan he the 41 trong cd tinh chat
B6 de 1. Neu
..., air) > (b p ,.., thi (05 + c 1, + c.) > (b 1 ..., bn + cn) vet most'c3, ..., en) E Nn .
Chung mink. VI
(ct i , ..., an) > (b i ,•bn)
ngn cd mOt chi s6 i = 1, sao cho
a t 1 b a. = b. 1-1. , a > b. 1 do dd at + ci = b i + c t , + c1_ 1 = a1 + c i > b + Hg quit. Neu , an) > , tin) uct , cn) > (d i , , de) thi (a1 + + cn) > (b +d1, ..., b n + dn)
Clang mirth. Theo bd dg 1
(ai + cp ..., an + en) > (a l + b„ + cn) >(1), + dt by, dii). n
Tit 136 de 1 va he qua cua no, ta say ra
Eljnh H 2. Gin set f(xi, xn) vh g(xi, xn) lit hai da auk khan
0 caa vanh tit] c6 Mc hang to cao nhdt theo thd tit lit
clxjii .. xnm oh d ixin NM c i d] x 0 Oa hang tit cao nhdt eta da that tich •••, xn) x,,) la cid ixlii +ba
Chang minh. Gia su
x n) = c itin + + ettain „,x:e
g(x i ,..., x n) = rbn in cipreirnt x:ron
da Ogee sap xgp theo 161 to dign. Digu do c6 nghia la aln) > (an ,..., aid veil mai i = 1
va (b 11 ,..., b in) > birit val moi j
Ta hay chang minh
d I . tam + b1
la hang to cao nhgt cua da thee tich
xn) xn).
Nhan f(x j ,..., xn ) vdi xn) ta &roc
xn ) = di + ... +bjn
ij
i= 1, ..., 1
j = 1, m
Mai hang to cs xla .,. x:e +bln cho ta pagn tv
÷ bJ1r” ath + bird E Nn
Theo bd de 1 Ira he qua dm nd, ta c6 rat bgt Bang thin
+ bin) > (
bin) > (aa + b ii ,
a bt bin) > (a11 + as, by) ,
Vay hang tit
Ci di aqii + b 11 + b
chinh la hang W cao nhgt e8a da thdc tich. n
HO qua. Ndu A la ntbt miM nguyen thi xn1 cling bay.