- X6thdms6 theo bii5n mcri voi diAu kiQn cua bi6n moi vtan6u k6n, tutl6 tatimdu_o c t6t tu6n cho bdi tor{n.
Tird6 maxD =7 datduockhi a=b=c=1 A 6',
,')12 Xdthirm s6 /(1)= 3(l+r)+ l+,, ff6nte(0: ll:c6 1rlr-1\/15 -1\
fol=ffi>0, v/e(o; 1) n€nhirms6 /(r)
ddngbi6nft6n (0; 11. Suyra mix l|=f()=+.
re10: ll o
\
Tird6 maxD =7-. datduockhi a=b=c=1. A6', 6',
2.2.Dinhgi6 hai bi6n d6i x*ngVdi mUc dich dua vC x6t hdm m6t bi6n :
2.2.Dinhgi6 hai bi6n d6i x*ngVdi mUc dich dua vC x6t hdm m6t bi6n :
bi6u thirc hai bi6n c6 vai trd gi6ng nhau thi chring ta
c6.th6 sri dgng c6c BDT quen.thuQc d.i d6nh gi6 hai bi6n tl6. Ket hqp vcri gi6 thi6t rdng bu6c gita c6c
bi6n, chirng. ta dua tlugc bi,3u thric dd cho vA bii5u
thirc mot bien.
. 5hi gap bi6u thric ba bi6n.s6 md trong d6 chua c6
bi6u thric hai bi6n vai trd gi6ng nhau, ta phii d[t 6n
phqr chuyiin bdi to6n vA c6ch vria n6u 6 tr6n.
Thi dU 5. Tin gia tri nh6 nhdt ct)a bi€u thti't-
_ u2 4b2 36t.3
E=---++-:::.. trons do
| + u)t (l + 2h)- (l + 3r')r
a, b, t' ld cac sij thqrc clro'ng thou mdn ot , = l.o 1 1 I ..
Loi gidi. D{t x=-:,}=--:-. z= - thi x. v. z>0a' 2b' 3c a' 2b' 3c
vd ryz:1. Lfc tl6 bi6u thric
o- I - | - 4
- - (l+xf - (l+r, - 3(l+z)3'
Ta thiy trong E c6 xu6t hi6n bi6u thric hai bii5n c6 vaitrd gi6ng nhau le ;l - +--!-, cho ta nghi
(l +x)'z (l +y)'
tli5n viQc sE sri dqrng BDT cho bi6u thirc ndy.
lll
Ta c6 RDT + > (*)(l +x), (l +y), I +.ry (l +x), (l +y), I +.ry
Ding thirc trong (*) x6y ra khi vd chi khi x=y:1.
Th0t v0y, BDT
(*) e ry(, - y)2 +(l- xy)2 > 0 1u6n dring.
Ap du"g (*), sri dpng xyz= 1 c6
l4z4
I- ' ' = - L-
l+xy 3(l+z)3 l+z 3([+z)3Xet hdm s(: Tet=Ti.#,trdn (o; +co). Xet hdm s(: Tet=Ti.#,trdn (o; +co).
t4'Ia co f '(:) 'Ia co f '(:) (l+z)2 ll+z)a f'(z)=0ez=1e(0; +o). Lfp b6ng bi6n thi€n hdm s6 th6y min f (z)= f 0=1. ze10; +o; J _(z-t)(z+3). (l+ z1a /(r) tr6n (0; + co), ) Tu d6, minE=i, 3 bi€u thu'c x,y,z ld thu'c trong ntdn
itpt ttusc khi (a: u: O=( t:1, 1'l O
\'2 3)
Thi dg 6. Tim gid tri l6n nhdt cua
F=-:Ln--!--*--L tronc do
Jl +.r2 Jl I y' Jl +:
cacsiithlrcth6amdn 0<x< y<z<4 vd xyz=1.
Ldi gidi. Ta c6 x,y,z>O vir l!z<4 suy ra
1
xy=:<l. Ap dung BDT Bunyakovsky vd BDT
z
l+a2 l+b2 - l+ab
clqc t.u chirng minh). Ta c6
( , * t )'.r( , * l ).4
[..nix, ,tl.V ) --l l* x2 t+y2 )- t**y
=l*=!=-!.noaoJl+x2 ,ll+y' JI+ry
2 t 2 t 2Ji+l ,11*rv JI+z /,, I "ll+z Jl+z \,, , - '> l.-t X6thdms6 112;=:# voi zell;41, c6 Jl+z t[- f ' ct = ---::::-, f ' (=) = 0 e z = 4. 2(1+ zl,,lz(l + z) ' Lpp b6ng thi6n hdm sO f (r) tr6n [1; 4], ta thdy max f(z)=f(q=Ji. Tt d6 maxF:Jj, dat ze1;41
(t r \
tluockhi (x: v: z\=11: '\22 i: 41. ) A
2.3. D{nh gi6 vi t<6t trqp A6i nii-{n cho mQt s6 biri
toin kh6ng c6 d4ng tlic biQt
Thf dg 7. Tim gid tr! nho nhat cira bi€u
^ .rr+-l 2+36 yr+r'rr36 2zt+22+9l:-_tL l:-_tL
2(x+l) 4(y-lS 2z+ldd x, !, z ld car: sij thwc daong thoa dd x, !, z ld car: sij thwc daong thoa
r - \2 (.r -llr -[ ,-]-l- .r, < 2. \.' ')) + ,r.1 TOAN HQC 5U'cruag@ 1899x2y2