S'(ct)
gi6 flt) th6ng qua hdm trung gian g(t). (ViQc chimg
minh v6i d6u I hodn todn tuong t.u v6i cr ld di6m cuc dai cira hAm s6).
THUTVAUN6DUNG
.'i,t:.. .-,,,, .,.,, il,i,
r . ,..r:' i, i,.i 'i:-
DANG THANIIHAI
(GIl'fHPf ,ti.t€u'Q14a7g Ph4c, YAn lg, Htmg YAn)
Bnbc 3z Xdt sqt biiSn thiAn cfiu hdm sA dgc trrng tr1n miin D. Th iili chrhng minh duqc bdi totin.
MQT SO \/i DU IrrNH HQA l. Dgng bdi totin thd hiQn sin ctic hdmflt) vd g(t)
Bii torln l.l. (MO - Hong Kong - 2005). Cho cdc
s6 a,b,c, d daong co tiing bdng l. Chung minh
I6(a] * h] +c] -d] 1>(.r: -h: -,1 - rl'l- ). 6(a] * h] +c] -d] 1>(.r: -h: -,1 - rl'l- ).
8
Phdn tich.D6u ":" xdy ra khi a=b=r=d =l4
I -.- -1 , I =c =1. 4. Xet ..-- ..-... hdm J f (x):6x' \.-| _ -' .o\--l x2 g(x)=x:61 =-' 4. .( r ) J I q) s Nen rn : "fI = , ,o d6 hdm h(t) c6'nxdt sE ld: " (+) he)=6r^ -,' -f voi r e (o :t ).
Ldi gidi.X6t hdm s6: fttl =613 -,' -f ffen (0 ; 1). Ta c6 h'(t)=ra2 -r,-*=o <+r=-f Goai);
f '(x't=r8x2 - ,r = ,'(i)=*,r'[+)= ,.1 1 t=-4 Ta c6 b6ng bi6n thi6n: t 0 1 4 I h'(t) -0+ h(t) ---_ _r v
Tri bdng bitin thi6n ta {lugc: htr)>-_-l '8 v6i
Vre(0,1), t&c lir 6t3 >i *1, _'1. Thay tlAn luqt
88, , ).. , , )..
boi a,b.c,d r6i cdng c6c ve lai ta duoc:
X6t g(t) : t, JU)
r ,( r )
=o=:. , Ir,
6(a3 +b3 +c3 +d3)>(o'+b2 +c2 +d2)
*|to* b+c+or-,f, =a2 +b2 +c2 +d2 ** o
Bii toin 1.2. (DH kh6i A - 2003) Cho x,y,z ld ba
sii daong th6a mdn: x + y + z i.I. Chung minh rdng:
Phdn t1ch. P5, cc-rr
* = #= -f . ua- cAn x6t ld
htrt=JTi -,*f,t v6i re(o;3).
Ldi gidi. X6t hdm s6 ht,t=JTi _r+f r'vot
r e (0r3). Ta co: h'(t)=#.ir- I = 0 er=le(0:3). Ta c6 b6ng bi6n thi6n t 0 I J h'(t) 0 + h(t) 11 T
Do d6: nro = Jz + r - t *f;,'={!, ore (o;3),
4,1=f;o,=t=r>3.f; -*(f +u2 +c2):2.VQy minP = 3 dat dugc khi a: b : c : 1. tr VQy minP = 3 dat dugc khi a: b : c : 1. tr Bii tofn 1.4. Voi cdc s6 x, !, z dwctng thda mdn
,v +y + z <12. Tim gid tri l6n nhdt ct)a bi€u thac
D_ I 1 I
t-...--
' 8-vE 8-,[ 8-.E'
Hrtng ddn. Xdt him : ' I I
s0 /'(r):8J -Ti4, ,rcn
khoAng (O;tZ).r6,p bing bii5n thi6n ta suy ra:
5h(t)<l-. Khi d6 ta c6: h(t)<l-. Khi d6 ta c6: Jtl _r 3.5 Il5 I P<-(.r+\.+:)+-'- < - +'" - - - 144\" / -' 36 - t2 36 2' I
maxP = i, diu ":" xdy ra khi r : ! : z: 4. tr
z
BAr rAp cr)Nc DANG
1. Cho a,b,c duongth6a mfln 1*1*l=3. Chr.mg
abc: 2a+l 2b+l 2c+l 9 : 2a+l 2b+l 2c+l 9
mtnhmhft:I:,:\_
" ,l3a+l J3b+ I J3c +l - 2'
2. (Hoc vi€n BCVT - 2001) Chrmg minh rbng v6i
mgi a, b,cth6amdn a + b +c : I thi I I I ^(a b c\ _+ : +_>Jl _+ . +_ l.
3o 3o 3' (3', 3o 3" )
3. (MO - Rumania - 2005) Gi6 sir a, b, c ld cilc s6lf1') lf1')
' ( 3, 4oJB2
''= r't=- 4l 'SuYra:
'[aJ
ney: y(t1+4of s(r), re(o;t).
Ldi gidi.X6t hdm s6 i (/): I' I 4O'lS2
{/- +; +-fi-t v(rir e (0,1). Ta c6: r e (0,1). Ta c6:
t'rrt = i[. += o <> / = ] e (o;1).
Ta c6 bing bii5n thi6n
t 0 I ;J 1 h'(t) 0 + h(t) 27j82 7 41 Ttd6: nQl, !@-, Vre(o; l) h^y F*), -ooff , * lP ru,
r,"r(1)> -oof u+y+z)*"'if >Ju. n
Bii tofn 1.3. Cho cdc sd thwc du'ctng a, b, c th6a mdn'. a2 +b2 +c2 =3. Tim gid tri nh6 nhtit ctia bi€u thu'c: P : $ +
" +.[Z * +.Jli - 1a + b + c).Phdn t{ch. D6u ":" xiy ra khi a=b=c=l= Phdn t{ch. D6u ":" xiy ra khi a=b=c=l=
cr =1. Trong bdi ndy: g(t)= t2, f(t)=.{3 +t _t,
duong th6a mdn a+b+ c = 3 . Chrmg minh r6ng
l*1*1 > a' +b'+c2.
a' b' c'
4. Gi6 sir a, b,c lit ciic s6 ducmg c6 tich bing 1.
rll
Chims " minh -=r-*---:--*---> a'-a+l b'-b+l c'-c+1 l.
2. Dgng bdi torin phdi bi6n dfi ad xuiit hiQn hdm
J(t) vd g(t')
Vni d4ng bdi tQp ndy hgc sinh cAn Hnh hoat, s6ng t4o a6 ui6n d6i, danh gi6 ttugc BDT, dua vd dpng ttang x6t.
Biri toSn 2.1" Cho a, b, c ld coc' .s6 clrottg thoo ntdn
"[i,)tr kir;tt ul,r' l. ( hing mitth r,'itt,q.
tt lt , .]uT
rl-, Jl ,' vT-..- l
Phdn tich. Gia thi6t a, b, c ld c5c s6 ducrng, tich
abc: I sE du-o. c tlua vd <ling thfc dOi xrmg cria t6ng
bing c6ch l6y l6garit n€pe hai vi5 rdi Aat x : lna,
y : lnb, z : lnc.X6t hdm g(t) : t, 71tS=$.Jl+e' Jl+e' Ta c6: 3g(o)=0=ct=0, g'(0)=1 ; f,(t)=:A f,<otJ@,^=+. ' zJ(l+ e')' ot 30, Do tl6 ft(r): -:- : ,lti 8 "
Ldi gidi.Tir a, b, c duong, lAy l6garit n6pe hai vi5 ci;6 abc: I ta dugc hq + lnb +lnc : 0. D6t lna : a
lnb = y, lnc : z, ta c6 giit thilit: x, y, zeJR vit
x + ! * z:0. BDT dd cho tdthdnh: "-"-"r30 Jl+c .,lt.-a ' Jt+7 - 2 Xdthdmso l,(r): +JP t tnt Jl+ e' 8 e'' +2e' ^ t- h'(tt=:-!A=0<>t=0. 2,!{1+ e')r I
Ta c6 bing bi6n thi6n:
t;
Tri bing bi6n thi6n ta dusc: hU)>;, Vt e lR tuc
ht Lr*,*9. rhay r lin luqt bbix, v, z
Jl+e' 8 2
roi c6ng c6c v6 14i ta dugc:
e' eY - ^l=
----: * 7 * -L> r1z 1,r * r' + z)Jl + e' ,!l + eY .,11+ e' 6 Jl + e' ,!l + eY .,11+ e' 6
JA '2 -3O
2
Ddu*r' x6yra e x:y:z:0 hay q:b=c:l.J
Bii toin 2.2 . Gia st)'rt, lt,, lrt t,tr' ,,i,ltt''tt:: tltrt,r
tndn u - h',' - l. Chilng mirrlt t',itt!.
u I h * ' .>a rt,.
l. ltt l't'rt I -.r/'- l()
Phdn t[ch.Oi5 <lua vC dAng to6n dang x6t, ta su dgng
.- (x+v)2
drinh gi6 xy <ff,Vx,y >0. I(hi tl6:
vr1r1= ii.iA*i*.:i.n,*4bc bc -.-gl9t- r-glo)' 4 '' 2 _4b-4, a2 -2a+5 ' b2 -2b+5 ' c2 -2c+5'
Bdi to6n trd thdnh: Ybi a, b, c ld cdc si5 duong th6a mdn a+b+c =1, chimg minh ring:
= 4o
' = ob - = 4.' -=| rzi.o
a2 -2a+5 b2 -2b+5 c2 -2c+5 lo
NhQn xdt. ViQc chimg minh (2) lirc niy hi5t sirc ttcrn
gian. Bpn dgc hodn todn c6 th6 lem dugc theo phucmg ph6p da trinh bdy.
BAI TAP Ct]NGDANG
1. Cho a, b, c duong th6a m5n a2 +b2 +c2 =L.
Chimg minh ring
o - b - c ,3€
b'+c2 ' c'+a' a2 +b2 - 2 '
2. Cho a, b,c duongvi th6a mdn: a+ b +c =9. Tim
crNNcriaP: ##.#s*44
3. Cho a, b, c duong. Chfug minh ring
abc-9(b+c)2' (c + a)2' (a + b)2 - 4(a+ b+c)' (b+c)2' (c + a)2' (a + b)2 - 4(a+ b+c)' t -co 0 +co h'(t) 0 + h(t) J1 2 so'?!H,H8S '*
,'[+) =6; eQ):,; o=+ -m=r'(+)'r'(+)=6
4. Cho a, b, c ducrng th6a mdn a2 +b2 +c2 =3.
.l1l
Chrmsminhrdns: " " 4-ab 4-bc , ' , * ' + ' 4-ca <1.
3. Dgng todn vdri BDT thudn nhfu cl6ng bQc:
Bii to6n 3.1: (MO-USA-2003) Cho c6c sO duong a
b, c. Chtmgminh rlng:
(2a+b+c)2 (2b+a+c)2 l2c+b+a12
---;--- T---:--- T---;--- ro\t,,
2a' +1b+c)' 2b' +qa+c1' 2c' +(b+a)' PhAn fich. BDT le e5l xrmg thuin nhdt bdc 2 n6n c6 thC dit a + b + c: l, a,b,c e(0;1) khi d6 BDT c6
r__-_ (a+l)z (b+l)2 (c+ l)2/I^6^. '-- -' r ' ! /I^6^. '-- -' r ' ! -:__________L/ O 3a' -2a+l 3b' -2b+1 3c' -Zc+l 4a +l 4b+l 4c +l 2l 3a'-2a+l 3b'-2b+l 3c'-2c+l 2
o4t - lirp-_1l:|L- 3t' : s'1:y=-t?r'-6t+6.-2t +l (3t' -2t +l)' -2t +l (3t' -2t +l)'
vdy ham cin xdt la: h(r t = - ,4' ! |
- 6, .
3t' -2t +1
LN gidi. Xdt hdm ,o, n6=-J!!)--6,'
3t' -2r +1
r€(o; l). Ta co: ft'1r; - -l?t') -6r +9-6=o
(3r' -2t +l)'e-2t2 -t+1=9ta +4t2 +l*4t-12t3 +6t2 e-2t2 -t+1=9ta +4t2 +l*4t-12t3 +6t2
€3to _14tr +4t2 -r=0<+r=0(loai); r=1.
J
,./,
la co bang bren thlen :
t 0 ;J1 I
h'(t) + 0
h(t)
3
,
Tri bing bi6n thi6n ta duoc: h@<1, Vr e (0;1) hay
4t+l - 3..
--;--:---i- < 6t + i Vr e (0; l). Thay t lin luot b6i
3t' -2t +l 2
x, !, z .Ta ilugc VT(2) < 61x +y * I '2 *2= 1. 2 n
Ilii to:in 3.2" (.\'tO .lttltutr 19t)J) d'ho ctit .t,)
tluttn! ,i. 1,., . {'l11rrt.q t,ti;tlt t'ritt!
(--{r+t'+(,ir ( b+,-r+r')r ( r th-*rrl -. J,- ,r, - -' l, * -, t;;,- -' 5' ,- ,r, - -' l, * -, t;;,- -' 5'
PhAn fich. BDT d6i xrmg d6i voi a, b, c n6n c6 thti
gi6 sria + b + c = l; a,b,ce(0;1). Khi d6 BDT dA
cho trd thdnh: (t-2a)2 ll -2b)2 ( -2c)2 _ 3 o\l-of- b\(-X- d+(l-qt' z5 1 1 r ,27 H---;-T-.:--T-=-=:-j-" 2a2 -2a+l 2b2 -2b+1 2c2 -2c+l - 5 ' I l(t)= -- :s01=r: te(0:l): ' 2t' -2t +l ' I ",, . -4t+2 .,,(t) 54 _ a= ^:f'(t)= ; : :f'l =l=!== hdm 3'r \" ei -%+lf '' [r.,1
cen x6t tit h(t)=**-*,, k.rrirc sir h(t)
'1 ^ 54t +21tr6n(0-l)thi ft(r) <'' * f(t\<" " '-' tr6n(0-l)thi ft(r) <'' * f(t\<" " '-'
25 25
+ J@)+ J'@)+ f (c)r#O+b+cS+fl=!.25'-- " "' 2s s'
Tri tl6 ta d6 ddng suy ra ldi gidi cria bdi to6n niy.
Bii torln 3.3, C'ho tt,b.t'lit l,,r ,r,) rltr,ttrt:. ( hitrt,q ntittlt
: t/ /, , ()
(( r /)/ lr/ -r i (,/ /rr' -ltri - /, . t' PhAn fich. BDT d6i ximg v6i a, b, c ndnkhdng m6t
tinh t6ng qudt, giir sri a * b + c: l, a,b,ce(O;1). BDTtrothdnh: --i-+ b -* c->2. \l-a)' ll-b)' (l-c)' 4 t | ^. l+t Voi f(t)=, ;g(r)=Itcr.=;i f'1t7=--'::-' (l -r)' r (l -r)' ",(r) e e /'[ir=r=m=;' Him s6 cdn x6t sE ln
ng1 : --t - -]. rfrao s6t su bi6n thi6n cria ft(r)
(l -t)' 2
7 7 -9t 3 .,..
ta dugc h(t)>-+ hay . >=-= va loi giai
4 '0-t)' 2 4cira bdi to6n dd trd 16n don giin. cira bdi to6n dd trd 16n don giin.
BAr rAP cr)Nc D4NG
1. Cho a, b, c duong. Chrmg minh ring
a b c t(t I l\
-I-l->_r_r_!
|b2 +c2 ' b2 +c2 '
c, +a, ' a, +b, - 2lr' b' , )'
2. Cho a, b, c duong. Chr.mg minh rdng
ab bc ca a+b+c
_!_I_
b+c c+a a+h 2
3. Cho a, b, c duong. Chrmg minh ring
ao b4 co a+b+c
-I_!_
b3+c3 c'+at a3+b3- 2
ffi lqp*i lilltHg(uilUdl mi