V i vay dung phi/dng phap mien gia tri ham so' giai bai todn nay la each giai tif nhien va cung kha ddn gian!. Cach giai tren la sif phoi hitp kheo leo giffa hai phffdng phap mien gia ir
Trang 1Chuyen dg BDHSG Join g\i tr| Ifln nh^t va gia trj nh6 nhat - Phan Huy Khil
min P = min< min P; min P > (2)
(x;y)eD [(x; y)eD| (x; y)eD2 J '
Dc'n day gpi m la gia tri luy y cua ham so:
Cty TNHH MTV DWH Khang Vijt
p 0 i 6 Tim gia tri I6n nhat va nho nha't cua ham so:
f(x) = -3 + 4X ^ + 3 X ^
( l + x ^ ) '
vdi X e •
Hudiig dan giai
Goi m la gia trj tuy y cua ham so: f(x) = 3 + 4 x ^ + 3 x ^
Xet hai kha nang:
1 Neu m = 3, khi do (2) c6 dang x^ = 0, vay (2) c6 nghiem
Do do m = 3 la mot gi^ trj cua f(x)
2 Neu m ^ 3, khi do (2) c6 nghiem khi va chi khi phiTcfng trinh (m - 3)t^ + 2(m - 2)t + m - 3 = 0
c6 nghipm t > 0 Dieu do xay ra khi A'>0
= 1 > 0 , d day S va P tiTdng tfng la long va tich hai nghiem
Trang 2ChuySn dg BDHSG Join gia tr| I6n nhSt g\i tr| nh6 nha't - Phan Huy Khai
Co
Nhdn xet:
1 Ta da suf dung cac dinh l i ve dau tarn thuTc bac hai de tim dieu kien sao cho (3)
nghiem khong am < i - ,
2. Ta CO the giai bai toan tren bang nhieu each khac
- Bang phifdng phap chieu bie'n thien ham so' (xem bai 11, b ^ i 1, chiTdng 4
cuo'n sach nay)
- Bang phtfcfng phap lu'dng giac (xem bai 5, chtfdng 3 cuon sach nay)
Bang phiTdng phap bat dang thtfc (xem bai 5 phan nhan xet, chtfong 3
G p i m la gia tri tuy cua ham so': f(x) = x + ^/x^ + — , x > 0
K h i do he sau day (an x):
(1) c6 nghiem o m ' ' - 8 m > 0 o m > 2 (chu y i\i (2) c6 m > 0)
K h i m > 2, thi (3) c6 nghiem M a t khac ta c6: P = — > 0; S - — = — > 0
Me'u ta CO bai toan sau:
Chtfng minh rang vdi moi x > 0, thi f(x) = x + \\ + — > 2 ' " '
V X
Khi do bai toan ddn gian hPn han That vay:
• Ne'u X > 2, thi ro rang f(x) > 2 yj ,,
Tuy nhien vdi bai toan da cho, thi lam sao ma tim ra hang so' 2 Ci'i kho cua
bai toan la d cho do V i vay dung phi/dng phap mien gia tri ham so' giai bai todn nay la each giai tif nhien va cung kha ddn gian! ' '
Bai 8 Cho ham so f(x) = x + Vx-x^ •
T i m gia tri Idn nha't va nho nha't cua ham so tren mien xac dinh cua no
Tir do de dang suy ra hai dudng noi tren cat nhau khi va chi khi x + y = in
nam giiTa hai difdng x + y = ( ) v a x + y = • •
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
Trang 3Chuyen dg BDHSG To^n gia trj I6n nha't gia trj nh6 nhat - Phan Huy Khii
^ 1 Cach giai tren la sif phoi hitp kheo leo giffa hai phffdng phap mien gia iri
ham so \k phifc^ng phap do thj
2 Xet each giai khac sau day (bang phiTdng phap chieu bien thien ham so)
D 'AU cua f ' ( x ) la diiu cua 8x - 8x^ - 1
f * Nhir vSy dau cua f ' ( x ) khi ^ < x < 1 !a dau cua 8x - 8x^ - 1
Tff do ta CO bang bien thien sau:
Ta thu lai ket qua tren! '
| L a i xct each giai sau (bang phu'ctng phap Iffdng giac hoa)
I D o ( ) < x < 1 = > d a t x = sin'(p (0 < (p < ^ )
I T f f d 6 x + V x - x ^ =sin^(p + 7sin^(p-sin'*(p = sin^(p + -y/sin^ (p(l-cos^ (p)
= sin^cp + sin(pcos(p (do sincp > 0; cos(p > 0)
Cue- i»w/2 //?zc/j ciich liicii nao?
Bai 9 Cho x va y la cac so thiTc thoa man dieu kien (x^ y ' + 1)^ + 4x^y^ x^
T i m gia tri Idn nhat va nho nha't cua bicu thtfc P = x^ + y l
HUdng ddn giai
Goi m la gia tri tijy y ciia P tren mien (x; y) thoa man dieu kien cho tri/(?c
Nhir vay he sau day(an x, y )
\^+y^=m (1) (x^ - y^ +1)^ - 4x^y^ - x^ - y^ - 0 (2) CO nghiem
Trang 4Chuy§n dS BDHSG Toan gia trj Idn nhat va gia tri nhd nha't - Phan Huy Kh^i
3 - N / 5 3 + >/5 (4) (an X) CO nghiem <=> m" - 3m + 1 < 0 <=> < m < • (5)
2 ' 2
N h i r v a y ( 5 ) l a dieukicnde (4)c6nghicm *'
Ta CO (3) <=> 4x^+ 4y^ = 4m <=> -m^ + 3m - 1 + 4y^ = 4m <=> 4y^ = m^ + m + i
Do m^ + m + 1 > 0 Vm, nen lif do suy ra he (3) (4) c6 nghiem khi va chi khi
Nhan xet: Bay gid xet each giai sau day:
Theo iren dieu kien da cho v(3i (x; y) c6 dang
Ta thu lai ket qua tren Phep giai nay c6n ddn gian hdfn ph6p sur dung mi^"
gia tri ham so
Cty TNHH IVITV DVVH Khang Vi$t
Qlum^^ PHIfONG PHAP SO DUNG DO THIHOAC HlNH HOC
^ Dt TIM GIA TRj LON NHAT VA NHO NHAT CUA HAM SO
phiTrtng phap nay dSc biet thich help vdi nhffng bai toan tim gia trj Idn nhat, nho nhat cua ham so, trong do cac bicu ihifc va dieu kien cua bai toan ban dau da tiem an nhffng ycu to hinh hoc ma thoal lien ta chffa nhin ra no Vdi Idp bai loan nay, ta lhff(:(ng sff dung cac tinh chat hinh hoc sd cii'p sau day:
- Trong ta'l ca cac dffdng noi hai diem A, B cho trffdc Ihi dffdng thing noi A,
B la dffdng c6 do dai nho nhat ' ' • ^ n t l c ' l e
Trong mot tarn giac long hai canh luon luon Idn hdn canh Ihff 3
- Cho diem M d ngoai dffdng thang d cho Irffdc Khi do do dai dffdng vuong goc kc tff M xuong d ngan hdn moi dffdng xien ke iff M xuo'ng ciing dffdng thang d
- Trong cac lam giac cimg noi ticp mot dffdng Iron, thi lam giac deu la tarn giac CO chu vi vii dicn tich Idn nhat
Ngoai ra ngffdi la cung dung cac kicn ihffc sau day:
i ' Dung tinh chat ciia do dai vecld, hoac tinh cha't cua ti'ch v6 hffdng cua hai w-vecld , :, S « ^ V * 1 ertH>'('^"M*-MiM ^
Dfing phffdng Irinh dffdng va mtit va cac cong Ihffc tinh khoang each cua
I hinh hoc phSng va hinh hoc khong gian ,
"ai 1 Be thi tuyen sinh Dai hoc, Cao ddnf' kho'i A)
Cho X, y, z la cac so ihffc dffdng va thda man dieu kien x + y + z < 1 Tim gia tri nho nhat ciia bieu ihffc
N ' i tiid",} UtKi.l «J!i!'ft !f''.« , •
Theo tinh chat ve do dai cua vecld long, la c6
T f f ( l ) c 6 :
u + V + w > U + V + w .(1)
O/l 1
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
Trang 5Chuyen ai BDHSG To^n gia tr| Ifln nhft v i g\i tr| nh6 nhflt - Phan Huy Khii
(x + y + z) + ' I 1 — + — +
(2)
Dau bang trong (2) xay ra o u, v, w la cac vectd cung phiTdng, cung chieu
v = k 2 W , k 2 > 0 '1-.' :\iu>,ii int't ri^'^rhb ;:5iim) ,0,': 'mM' uw-i (3)
1 Trong bai trcn ta da kct hdp phiTdng phap suf dung vectd trong hinh hoc phang
phi/dng phap siir dung ba't dang thtfc de giai bai toan dat ra
2 Thay cho phiTdng phap bat dang thtfc ta c6 the sOr dung phiTdng phap chieU
bien thien ham so nhi/ sau: j
Cty TNHH MTV DWH Khang Vi?t
Vaytuf(lO) suyraminP= ^^min f(t) =
Ta thu lai kct qua trcn ^ .^^.si^'K^'^Ai^-piih h': • '
fiiii2.(De thi tuyen sinh Dai hoc, Caoddnfi khoi B) i tr ;
Cho x y la cac so thifc tiiy y Tim gia tri nho nhat cua bicu thtfc >
P = V( X - l ) ^ +y^ + ^{X + lf+y^ + |y - 2| , „
V; 1 HUifiig dan giai
Ta kct hdp phi^dng phap suT dung vectd va chieu bicn ihien ham so' de giai bai loan nay Xem Idi giai trong bai 13, §2, chi/dng 1 cuon sach nay
Bai 3 (De thi tuyen sink Dai hoc, Cao dang khoi B)
Cho ham so f(x) = x + yfi-x^ Tim gia trj Idn nhat va nho nhat cua ham so
nay trcn mien xac dinh cua no
Hiiifng dan giai
Xem Idi giai ket hdp giiJa phifdng phap do thi, hmh hoc va phifdng phap mien gia trj ham so de giai bai toan trcn trong bai toan 1, §1, chU'dng I cuon sach nay
4 Tim gia tri nho nhat cua ham so f(x) = Vx^ - x + 1 + Vx^ - V 3 x + 1, vdi
X e R
Hiidng dan giai j
Viet lai ham so' f(x) diTdi dang sau day
va
245
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
Trang 6Goi C„ = A B n Ox Ta c6 C„A + C B A B
Nhir vay ncu dat Xo = OC,, , thi l'{xn) = >/2
Nhif vay la eo min = <=:> x = Xd = 73 - 1
Nhqn xet: \ '
1 X|) C O the linh nhif sau: , c
DifcJng thang qua A B eo phlfdng trinh
y + ] X - 73
2 2 2 2
Cho y = 0 x„ + — = ~ =>xn= yfi - i n ! m-^X ^m nhui
2 2(1+ V 3 ) , •
2 R6 rang phu'rtng phap do ihj to ro day uy life eiia no qua RJi giai trcn
3 X c l C c i c h gi£u bang phifdng phap hinh hoc sau day ^
N c u X < 0, k h i do ta c6
| - { x ) - V x ^ - x + 1 + V x^ V 3 x + l , •.tjiri • • : ( * > Y
i " ( - x ) - V x ^ + X + 1 + Vx^ + N/3X + I " ( * * )
Do X < 0, ncn lit (*) va (**) suy ra IXx) > f ( - x )
V i the do chi quan l a m den m i n f ( x ) , ta chi can xet k h i x > 0 S ^
4 Cac ban Ihijr x c m eo the giai bai toan trcn bang each khac ma l a i dOn giiin
hdn hai each tren khong?
- x + 2 y - 8 < ( ) Bai 5 Cho x, y la cac so thiTc thoa man dieu k i c n < x + y + 2 > 0 ,1
M ( x ; y ) n o i tren chinh la lam giac A B C (kc ca ba canh), trong do
Trang 7Chuy6n BDHSG Toan giA trj I6n nhal yia In nh6 nhift - Phan Huy Khii
Binh luqn: Khong the c6 phiTdng phap khac hieu qua hdn phu'dng phap do thi
va hinh hoc gian bai toan nay
A^^rtjcef: X e t bai toan tiTdng ti/sau: ; , ,» :
Viet lai P diTdi dang: P = (x - 2)^ + (y - Af - 20
Gpi I la diem I = (2; 4), khi do P = M l ' - 20, d day M ( x ; y) thuoc tiir giac
B a i 6 Cho cac so thiTc x, y thoa man dieu kien: x^ + y^ + 16 = 8x + 6y
Tim gia trj Idn nha't va nho nha't ciia bieu thuTc P = 4x + 3y
* Htidiig ddn giai
V i e t lai dieu kien da cho diTdi dang (x - 4)^ + (y - 3)^ = 9 (1)
Tilf d6 suy ra cac diem M ( x ; y) thoa man (1) n^m tren diTdng tron tam ta'
diem 1(4; 3) va ban kinh R = 3
Cty TNHH MTV DWH Khang Vigt
Khi (x; y) thoa man (1), ta viet lai bieu thi?c P drfdi dang
(2)
8 x ^ ^ x _ ^ y _ ^ ^ 3 ^ l , ,
- 2 2 2 ^
rkng neu ve he true tpa do Oxy, ^
d day M ( x ; y) thoa man (1) • TCf (2) suy ra maxP = 8 + - m a x O M ^ ,
Cac ban c6 the tU" nghiem lai cac ket qua (*) va (**) mot each de dang
^Hn xet: Ta co the S I J T dung phiTdng phap lifdng giac hoa de giai bai toan tren
Tiif do P = 4x + 3y = 4(4 + 3sincp) + 3(3 + 3 coscp) = 25 + 12sin(p + 9cos9 (3)
Ap dung ba't d^ng thufc quen bie't: V a , ta c6:
~-\/a^ + b^ < a s i n a + b c o s a < Va^ + b^
T a c 6 : - 1 5 < 12sin(p + 9cos(p< 15 ^"^^
*
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
Trang 8Cliuyen BDHSG To^n gia tri I6n nha't va gia trj nhd nha't - Phan Huy KhSi
Bay gic( liT (3) (4) suy ra maxP = 40; minP = 10
Ta thu lai ket qua Ucn (vdi phep giai raft gpn gang)
Bai 7 Cho x va y la cac so' thifc thoa man dieu kien •
Tim gia tri Idn nha't cua bieu thiJc P = x + 2y
1^-t)
i' l i t ; , i , ,
Cty TMHH MTV DWH Khang Vlgt
Gia su" M(x; y) la diem, trong do x, y thoa man he dieu kien da cho
De dang tha'y rang khi do mien D cac diem M(x; y) nhu" vay du"dc bieu dicn
tren mat phang toa do xOy la toan tarn giac ABC, vdi cac dinh A(4; 2);
B ( 2 ; 4 ) v a C ( l ; 3 )
Bai loan da cho trcf thanh: '•,
- Tim gia trj Idn nha't va nho nha't cua tham so m, sao cho diTdng thang x + 2y
= m cat tam giac ABC Bang each tinh tien difdng thang x + 2y = 0 (theo
chieumuiten) f m « ' l - - - x 1
•' Dieu dau tien ma difrfng thang x + 2y = m gap tam giac ABC chinh la die'm
Khi do m = 1 + 2.3 = 7 Diem cuo'i ciing ma du'dng x + 2y = m gap
AAB*-chinh la diem B(4; 2) Khi do m = 2 + 2.4 = 10
Vay maxP =10<=>x = 2 ; y = 4; min P = 7 < = > x = l ; y = 3
xet: Ta c6 the giai bai toan tren bang phiTdng phap mien gia trj ham so
nhu^sau:
: ,•„ ,.v,.;;,r ; ^ ^'.^^^'.1 i fX + 2y = m (1)
Goi m la gia tri tiiy y cua P, khi do he ,sau day (an x; y) • ^ ^ "^^ 10 > 0 (2)
x + y - 6 < 0 (3) ,^ [ x - y + 2 > (4)
Tir do suy ra maxP = 10 o x = 2; y = 4; minP = 7 o x = : l ; y = 3
^^nh luqn: R6 rang phuTcfng phap giai bang do thi va hinh hoc to ro hieu qua
hcfn h^n so vdi phifdng phap mien gia tri ham so j
^ ^ i 8 Cho X, y la cac so' thiTc thoa man dieu kien sinx + siny = ^
Tim gia tri Idn nha't va nho nha't cua bieu thtfc P = cos2x + cos2y
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
Trang 9Chuyen ai BDHSG Toan gia tr| I6n nhS't vi g\& tr| nh6 nhSit - Phan Huy Khai
Hiidng ddn gidi
Dat u = sinx; v = siny
Khi do ta c6 cos2x + cos2y = 1 - 2sin ^x + 1 -2sinV = 2 - 2 (u^ + v^)
Bai loan da cho trd thanh: \'y A-.:.,-':! \
Tim gia tri Idn nha't va nho nha't cua bieu thiJc Q = +
Ve he toa d p Ouv Khi do tap cac diem (u; v) thoa man (2) chinh la doan
thang AB (do la phan du'dng thang u + v = ^ nam trong hinh vuong canh 2
trong hinh ve dU'di day)
min OM^ -OH^ =
MeAB 8
(d day H 1^ hinh chicu cua O ircn AB)
Tiirdd suy ra:
Cty TNHH MTV DWH Khang Vi§t
xet: Ta c6 the suT dung phuTdng phap "chicu bien Ihien ham so " de giai bai
toan tren nhU'sau:
TH do suy ra max? = max f(u); minP = min f(u) vdi l(u) = -4u^ + 2u + 3 -^<u<l <u<2
Ta CO f'(u) = -8u + 2, va c6 bang bien thicn sau 'il '::i
Ta thu lai ke't qua trcn Phep giai cung gpn gang va ddn gian!
fiaf 9 Cho bon so thi/c x, y, z t thoa man dieu kien x + y = 6; + t^ = 1
""im g'la tri nho nha't cua bieu thtfc P = x' + y^ - 2xz - 2yt
Htidngddngiai t, , ,
Xet he true toa do Oxy Khi do diem M(x; y) vdi x + y = 6 n^m trcn dir5ng
ih^np X + y = 6; con diem N(z; t) vdi r + t^ = 1 nlm trcn diTiIng tron ddn vi
A + y^ =; 1 tam tai O va ban kinh bhng I
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
Trang 10Chuyen dg BDHSG T(rfn gi^ tr| I6n nhtft g\i trj nh6 nhift - Phan Huy Khi\
Vie'l lai bieu tMc P diTdi dang
Nhgn xet: LcJi gitii bkng phifdng phap do tfii that trong sang va gon gang!
Scr di CO Idi giai nhif vay, vi trong bai toan cai hon "hinh hoc" da the hicn 10
qua cdc dieu kicn x + y = 6; + t^ = 1 vii bieu thtfc P gdi ta ngliT den khoiing
each giffa hai diem tren mat phang tpa do
Bai 10 Cho bon so thiTc x, y, z, t thoa man dieu kien x + y + z + t = 1
T i m gia tri nho nhat cua bieu thiJc P = 7x^ +z^ + •
• • - VK!; f^^^,^g giai ' s^' ^-^i"
Tr6n mat phang toa dp xOy xet hai diem M(x; z) va N(x + y; z + t)
Do (x + y) + (z + t) = 1, nen diem N nhm tren diTdng thang x + y = 1
(3) (4)
T a c d O M = Vx^+z^ , M N = ^(x + y - x ) ' + ( z + t - z ) ^ =^|y^+l^ • '
Ta CO O M + M N > O N > O H (1)
Cty TNHH MTV DWH Khang Vi^t
d day H la hinh chie'u cua O
tren dtfdng thang X + y = 1
72 ' ^'^A
Ro rang do O H = — , nen ta c6 t i r ( l )
P = V x ^ + z 2 + V y ' + t 2 > ^ (2) * Pafu b^ng trong (2) xay ra
c> O M , M N cung phu-png, cung chicu va N = H
^Hn xet: Phu-png phap do thj to ro hieu qua trong viec giai bai loan tren Cac
hn c6 the giai each khac lai hay hdn each giai tren khong?
'^^J11. Cho X, y, z la ba so thifc dtfdng va thoa man dieu kien: xyz(x + y + z) = 1
Tim gia trj nho nhat cua bieu thiJc: P = (x + y)(x + z) OB,
* t Hudng ddn giai
^ t n tam giac A B C c6 3 canh A
'^B = X + y; A C = X + z; BC = y + z
^6 rang AABC ton tai v i thoa
"lan cac tien dc vc dp dai canh
^"Ja mot tam giac
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
Trang 11Chuy6n ai BDHSG Toan gia tri lan nhS't va gia tri nhd nhat - Phan Huy KhSi
Khi do ne'u goi p la niira c h u vi cua tam giac thi p = x + y + z
Thco cong IhuTc Heron, la c6
SABC = V P ( P - B C ) ( P - A C ) ( P - A B ) = 7(x + y + z)xyz
T i r g i a l h i e t s u y r a S A B c = 1- (1)
Matkhac SABC = - A B A C s i n A = j ( x + y)(x + z ) s i n A ' ' ' (2)
Do s i n A < l , n e n t i r ( l ) ( 2 ) suy ra ^ ( x + y)(x + z) > 1 => (x + y)(x + z) > 2
Bai 12 Cho bon so x, y, z, t thoa man dieu kien x + 2y = 9; z + 2t = 4
T i m gia tri nho nhat cua bieu thiJc
Tren mat phang toa do Oxy ve hai diTdng thang x + 2y = 9; x + 2y = 4
X€i hai diem
ginh lu4n: Khong the c6 IcJi giai nao lai ngan gon va hay hOn lc:fi giai tren
pai 13 Cho X, y la hai so thi/c thoa man dieu kien: x - 2y + 2 = 0
Tim gia trj nho nhat cua bieu thiJc
quen bie't ciia hmh hoc phang, thi neu gpi A ' la diem doi xi^ng cua A qua dirdng thang x - 2y +
2 = 0 va gia sur A A '
ca't dirdng t h i n g X
-2y + 2 = 0 tai M„, thi M A + M B > M„ + M„B, V M e (d), cl day (d): x - 2y + 2 = 0 , ,
Co the tinh diTdc ngay Mo = (5; 3,5) => MoA + MoB = A ' B = 6
Trang 12Bai loan trd thanh: \ rskmisl
Tim gia tri nho nhat cua ham so f(y) = 75y^-30y + 50 + 7 5 y ^ - 4 2 y + 98
vdi y e R
Cac ban hay thuT dung phifcfng phap chicu bien thicn ham so de giai tiep!
Sau do hay so sanh vdi each giai bang phi/dng phap suT dung do thi va
hinh hoe c( trcn! "•• a r i e b'iblJb H'l^si
2 Ta CO bai toan tiTdng ti/ (giai bang each sijf dung phi/dng trinh difdng thang
va mat phdng trong khong gian) ^ ^ ^ ^ , ^ ^ ^^^^^^ ^^^^^.^^^ ^
Cho X, y, z la ba so thiTc thoa man dieu ki6n 2x - y + z + 1 = 0 ^
<f Tim gia In nho nhai eija bieu thiJc:
HUofng dan giai
Vict hji l(x) di/di dang
Dc thay ABC la tam giac deu vdi tam la go'c toa do O *' Theo ket qua quen bict trong hinh hoc phang:
Do ABC la da giac deu, ncn vdi moi diem trcn mat phang toa dp - ndi ricMig vdi mpi diem T(x; x), ta cd
TA + TB + TC > OA + OB + OC
Do OA = OB = OC = 1, nen liT (2) (3) cd f(x) > 3 V x e R, >< /;?
f(x) = 3c=> T = O <=> X = 0 nfi^h'iit<ifi\;ii:m<x\ii\ xi\n i B % r f : j i i l
Bai 15 Cho x, y, z la cac so thiTc thupe doan [0; 1 ]
Tim gia tri Idn nhat ciia bieu thiye: P = x + y + z xy - yz - jzx, , ^ ij/p
HiUing d&n giai
Vc lam giac deu ABC canh bang 1 Tren cac canh AB, BC, CA Ian li/dt lay cac diem M , N , P sao cho A M = x; BN = y; CP = z > » r
Ta cd: S^MP + + -VNH ^ ^KMC • ( D Da'u bang trong (1) xay ra khi va chi khi
trong cac tam giac AMP, BMN, CNP cd tam giac trting vdi AABC Dieu dd xay
ra khi va chi khi Irong ba so' x, y, z cd mpt so bang 1, mot so bang 0 va mpl so
Trang 13Dau bang irong {**) xay ra <=>
Chuy6n 6i BDHSG Join gii tri Idn nha't va gia tri nh6 nhS'l - Phan Huy KhSi
TCf do suy ra maxP = I C5> x y, z ihoa man (**"•)
Ta thu lai kcl qua trcn , (
Cach giai niiy ciing ra'l gon gang va di<n giiin!
2 Ta CO bai loan lifting lif sau: Cho \ y / I ia bon so thifc ihuoc doan [0; 1 [
Tim gia Irj Idn nhat ciia bicu ihi'fc I ' \ y + z + i - xy - yz - zt - tx
Hifi'liifi dan fiidi
Ta giai bhng phu'dng phap sif dung hinh hoc nhU' sau:
Trcn bon canh cua hinh vuong: dal AM = x; DN = y; CP = z; BQ = I (xcni
NhiT vay maxP = 2
Cac ban hay tif lam hai dicu sau day:
- Khi nao ihi maxP = 2
- Giai lai bai loan trcn bang phu'dng phap "bat dang thu'c"!
3 Ta CO bai loan sau di/a vao bai loan da cho:
Cho X, y, z la ba so ihiTc ihuoc doan fO: 1 ]
Tim gia trj Idn nhat ciia bicu thiJc P = x^ + y"' + z^ - x'y - y^z - z^x
Lcfi giai nhUsau:
va P = 1 khi va chi khi trong ba so' x, y, z c6 mot so' bang 1, mot so' bang
0, so con lai tiay y e [0; 1] (ban doc co the nghiem lai dieu nay)
Vay maxP =1
pal 16 Cho X, y, z la ba so' thifc thoa man dieu kien x + 2y + 3z = 4
Tim gia tri nho nhat cua bieu thuTc: P = Vl6 + x^ + 2-^16 +y^ + 3 \ / l 6 + ?
Hudng dan giai
vdi X + 2y + 3z = 4, tren mat phang tpa dp Oxy xet cac diem
A(4; x); B(12; x + 2y); C(24; x + 2y + 3z). if r K)A d i - / i
17 Tim gia Iri nho nhat cua ham so f(x) = Vx^ - 2 x + 2 + Vx^ + 4x + 8 trcn
mien xac dinh cua no
Hudng ddn giai
R6 rang ta c6 x^ - 2x + 2 = (x - 1)^ + 1 > 0 Vx,
+ 4x + 8 = (X + 2)2 +4 > 0 Vx
Vay f(x) xdc dinh tren R ,f \
Viet lai f(x) durdi dang sau day
261
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
Trang 14Chuygn dg BUHSG Toan gia trj I6n nha't va gia Iri nhd nhfl't - Phan Huy Khai
f(x) = V(x-l)^+l^ +7l'^-(-2)P+(-2)- ( 1 ) ' - "
T r c n mat phang toa do Oxy x c t a i c d i e m A ( x - 1; 1) va B ( x + 2; -2)
T i r ( i ) c 6 f ( x ) = O A + O B • ^ ^ ^ ' ^ ' ' ^ ' ^ - ^ ^ J ' ^ ^ ' ^ i ^ ^ * " " ' " y ':<^^^^^; :
R6 rang ta luon c6 O A + OB > A B = sjy + 3' = 3 V2 ^ , ,
Vay Rx) > 3V2 V X e R :l,:o;> oofo m^v ( 3 ) - V M J u.i (v'v
Dan hang trong (3) x a y ra i %nim v;':
o A, O, B thang hang S «grtt cnrfi ni>f<f & ft{ x ,x osf
Nhgn xet: Ro rang phi/dng phap do thi to ro qua hicu qua trong hai b a i toiin 16
va 17 Cac ban c6 each g i a i nao khac ngan gon hdn khong?
Cty TNHH MTV DVVH Khang Vi$t
P^^Mif^ liNG DUNG CUA GIA TBI LlN NHAT VA GIA TRj
NHD NHAT TRDNG BAI TDAN GIAI PHU0NG TBINH
VA BAT PHUONG TBlNH CO THAM SO
dung gia t r i Idn nhat va nho nhaft cua ham so' la mot trong nhiyng phu'dng phap hffu h i c u dc giai phifdng trinh va bat phufdng trinh co tham so' Dang toan nay cung thi/dng xuyen xua't hicn trong cac dc thi Toan cua cac de thi tuyen sinh vao D a i hoc, Cao dang trong nhijfng nam gan day
§ 1 M O I L I E N H E G I Q A G I A TR! L 6 N NHAT, N H O NHAT C U A
H A M SO V A Si; C O N G H I E M C U A P H U O N G TRINH V A BA'T
PHUdNGTRiNH u
Ta thu'dng x u y e n sii" dung ket qua sau day:
Gia su" f ( x ) la ham so l i e n tuc trcn m i e n D , va gia suf ton tai M - m a x f ( x ) ;
xeD
m = m i n f ( x ) _ , 3-i;sjq|> ^a.ji,} ylvtfriH $ (MUV'
^, , f ijrf'i \n iuI i l n j m ^ k i s i i ' j •
Kin do ta co:
1 He phiTdng trinh r(x) = a CO nghiera khi va chi khi m < a < M
2 He baft phuTdng trinh \ nghiem khi va chi khi M > a
!• Gia suf he phufdng trinh da cho co nghiem, tufc la ton tai Xd e D sao cho f(X(i) = a
Theo dinh nghla ta co m i n f ( x ) < f(X(,) < max f(x), tiJc la
K N o i rieng no nhan gia tri a, tiJc la ton tai Xo e D sao cho f(xo) = a
1 Dieu do co nghla la phu'dng trinh da cho co nghiem tren D => dpcm
-• 263
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
Trang 15Chuy6n dg BOHSG Toati gia tii Idn nha't va gia trj nho nha't - Phan Huy KhJIi
2 Gia sur he da cho c6 n g h i c m , ttfc la ion tai x„ e D sao cho f(X(,) > a
3 Gia sur m > a • ,J'AHfl MOJ M A W Aiil© tSH H : M 'tOM
Lay X(, luy y e D Ta c6 f(x„) > min f(x) = m > a -Q ^4
Vay IXx) > a Vx e D HWmi
miOim-Dao lai gia sur f(x) > a Vx G D %(M£:9,ii&fU,
' ' K h i do do m = min f ( x ) , nen theo djnh nghTa ta c6 ton tai Xo e D ma m = f(x
x e D
Ttr f(X|,) > a => m > a Nhu^vay ta CO dpcm. , , 4 „ U > l j i w i i
4,5 Chu-ng minh tu-cfng tu" nhu" 2,3 I
, ' :o'} iiJ dh '
VA BAT PHOCfNG TRINH CO THAM SO CO MAT TRONG
CAC Kl THI TUYEN SINHVAO DAI HQC, CAO DANG
Bai 1 (De thi tuyen sink Cao ddtifi khdi A, B-2011) ' ' *'
Cho phiTdng trinh 6 + x + 2 V ( 4 - x ) ( 2 x - 2 ) = m + 4 f V 4 - x + V 2 x - 2 )
T i m m de phU'dng trinh c6 nghiem
265
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/