Chuyên đề bồi dưỡng HSG GTNN GTLN(Thầy phan huy khải)

IJCU N6I i>Au Bai todn tim gid tri U 'fn nhd 't, nhd nhd't ciia ham so noi rieng vd hat dang thiic ndi chung Id mot trong nhifng chii de quan trong vd hu'p dSn tnmg chutfng trinh gidn

0+150^7^ P.GS - IS PHAN HUY KHiH

Chuqen de

BOI DUONG HOC SINH

Gia tri I0n nliiK Gia tri nli6 nlid

^Danh clio hoc sinh Idp

Trinh bay bia : C o n g ty K H A N G V I E T

Chiu trdch nhi^m ngi dung vd ban quyen

Cong ty TNHH MTV DjCH Vy VAN HOA KHANG VI^T

SACH LIEN KET

CHUYEN DE BOI D J O N G HQC SINH G 1 6 1 GIA TRI LdN NHAT,

GIA TRj NHO NHAT

Ma so : 1 L-31 7DH2012

So lugfng in 2000 Wn, kho 16x24 cm

In tai Cty TNHH MTV in an MAI THjNH DL/C

Dja chl: 71 Kha Van Can, P.Hiep Binh Chanh, Q.Thu Dufc, Tp.HCM

So xuat bin: 1297-2012/CXB/08-213/DHQGHN, ngay 26 thang 10 nam 2012

Quyet djnh xuat bin so: 311 LK-TN/QD-NXBDHQGHN

in xong va n6p liAi chieu qui I nam 2013

IJCU N6I i>Au

Bai todn tim gid tri U 'fn nhd 't, nhd nhd't ciia ham so noi rieng vd hat dang thiic ndi

chung Id mot trong nhifng chii de quan trong vd hu'p dSn tnmg chutfng trinh gidng day vd

hoc tap In) mon Todn d nhd trudng phd thong. Trong cdc de thi mon Todn ciia cdc ki thi

vdo Dai hoc, Cao dang 10 nam gun day (2002 - 2011) cdc hdi todn lien quan den vi ^c

tim gid tri nhd't, nhd nhd't ciia hdm w thudng xuyen cd mgt vd thut'fng Id mot trong nhiing cdu kho nhd't ciia de thi , , , ,

Vdi li do do cdc cud'n sdch chuyen khdo ve chii de nay ludn luon thu hut su chii y vd

quan tdm ciia ban doc. Tnmg cud'n sdch "Cdc phUtfng phdp gidi todn gid tr\ nhd't, gid tri nho nhd't" nay, chiing toi se cung cap cho ban doc nhvtng cdch gidi thong dung nhd't doi vdi nhiing hdi todn tim gid tri Idn nhd't vd nhd nhd't ciia hdm so.cdng nhu hiet cdch dp dung hdi todn nay de gidi nhieu hdi todn lien quan den no

Noi dung ciia cud'n sdch dUOc trinh hdy trong chUcfng

Chiiong 1 v(H tieu de " Vdi bdi todn md ddu ve gid tri l^n nhd't va nhd nhd't cua ham so" se gidi thi^u vdi ban doc bdi todn tim gid tri Idn nhd't, nhd nhd't ciia hdm sd'thong

qua vi ^c trinh hdy tinh da dang ciia cdc phUcfng phdp gidi hdi todn ndy. Bdng cdch diem lai nhiing sU cd m$t ciia cdc hdi thi ve chii de ndy cd mdt trong cdc ki thi tuyen nnh Dai

hoc - Cao dang cdc ndm tic 2002 den 2011, cdc ban se thd'y duac sU can thie't cua vi$c

phdi trang hi cho minh nhvtng kien thiic de gidi quyet cdc hdi todn d'y Cud 'i chUtfng 1 Id cit sd li thuyet ciia hdi todn tim gid tri Idn nhd't vd nhd nhd't ciia hdm so. Phun nay giup cdc ban nhiing kien thiic chud'n hi can hiet di' doc tiep cdc chUifng sau ciia cud'n sdch Cdc phUcfng phdp ca ban vd thong dung nhd't de gidi bdi todn tim gid tri Idn nhd't vd nhd nhdt ciia hdm sd'duac trinh hdy tit chUOng 2 den chuang 6 ., •

Chitang 2: Phi/mg phdp h&t ding thuCc tim gid tri l^n nhdt vd nho nhdt cua ham sd ChiiOng 3: Phiicfng phdp liifng gidc hoa tim gid tri l^n nhdt vd nho nhdt cua hdm so'

Chitang 4: PhiiOng phdp chieu bien thien hdm sd tim gid tri Idn nhdt vd nhd nhdt cua hdm sd

ChiMng 5: Phiicfng phdp mien gid tri hdm sd tim gid tri Idn nhdt vd nhd nhdt cua hdmsd

ChUOng 6: PhUmg phdp dS thi vd hinh hgc tim gid tri Idn nhdt vd nhd nhdt cua hdm sd,

d mSi chuang, chung toi cdgdng truyen tai den ban doc n^i dung co ban cua phuc/ng

phdp, dUa ra cdc Idp hdi todn md phuc /ng phdp gidi no la thich h (fp nhd't. Thdng qua vifc

Phdn tich, hinh luqn vd dUa ra lam doi chiing nhieu phUtfng phdp khdc nhau gidi cUng

mQt bdi todn se giup cdc ban tim duoc cho minh mQt phuang phdp m vi $t nhdt de gidi hdi todn gdp phdi. Do Id dieu mdi me cua cud'n sdch ndy. Chung toi ludn ludn gia tinh thdn chii dao d'y trong tvCng phdn ciia cud'n sdch

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ChMng 7 danh de trinh hay vi$c ling dung ciia hai todn tint gid tri U'fn nhd't, nhd nhat

trong vi^c hi$n ludn phu<mg trinh vd hat phU(mg trinh co tham so Xin luu y v('/i cdc hgn

rdng day cdng Id mot chii de thi/dtng xuyen xud't hi^n trong cdc de thi tuyen sink vao Dai

hoc - Cao dang nhQng nam gdn day (2002 - 2011)

Phdn ddu ciia chiMng 8 vc'fi tieu de "M$t sobai todn khdc tint gid tri "hat vd nhd

nhd't cua ham so" de cyp den hai todn tim gid trf Idn nhat vd nhd nhd't ciia ho ham so

phu thuQC tham so

Cudn sdch nay chu yeu trinh hay cdc hai todn tim gid tri nhat, nhd nhcft trong Dai

.so vd Gidi tich

Bdi todn tim gid trf U'tn nhd't, nhd nhd't trong w hoc, hinh hoc to h(tp, hinh hoc khong

gian, hinh hoc phdng, luang gidc, se duoc chung toi trinh hay trong mot cudn chuyen

khdo khdc (sdp xud't hdn) Tuy nhien trong phdn hai ciia chU(fng 8, chung toi van ddnh

mot it trang de diem qua mot id thi du tieu hieu ddc sdc ciia cdc hai todn nay

Chung toi thiet nght cudn sdch nay se ddp dng dUilc mot sd lUOng Win hqn doc Cdc

ban hoc sink phd thong, cdc thdy to gido day Todn deu cd the tim dU(/c cho minh nhCtng

dieu hd ich khi doc no

Mat ddu vc'fi tinh than nghiem tiic, ddy trdch nhi(m khi viet cudn sdch nhung vdi mot

khdi lU(/ng U'fn cdn truyen tdi, cudn sdch khdng the trdnh khdi cdc khiem khuye't

Tdc gid rat vui long neu nhdn ditifc su gdp y ciia hgn doc, nhd't la cdc hgn ddng

nghi$p xa gdn de quyen sdch tdt hifn nQa trong cdc idn tdi hdn tiep theo (vi chiing toi

nght rdng chdc chdn cudn sdch nay ton duc/c tdi hdn nhieu idn)

Thtf tCf gop y xin guTi ve theo dia chi sau:

- P H A N H U Y K H A I ,

Vipn Toan hoc, 18 DiTcfng Hoang Quoc Vi^t - Quan C a u Giay - Ha Noi

Xin chan thanh cam dn

§ 1 VAIBAITOANMdDAU Trong m u c nay chung toi gidi thieu vai bai toan v e gia tri Idn nha't va nho nha't ciia h a m so T h o n g qua nhffng hai toan nay, chung toi muon de cap den c a c phtfdng phap c d ban nhat de giai c a c bai loan v e gia trj Idn nhat v a nho nhat se di/dc trmh bay k y y o n g cuon s a c h nay

B a i t o a n 1: (De thi tuyen sink Dgi hoc, Cao dang khdi B) '

C h o h a m so' y = x + V 4 - x ^ T i m g i a trj Idn nhat v a nho nhat c u a h a m so'

n a y tren m i e n x a c djnh c u a no

Hildiig dan giai !

Cflc/i 7 ; (PhU'dng phap bat dang thtfc) ' s

H a m so' da cho x a c dinh khi -2 < x < 2 j > ;

1 C a c h giai tren hoan toan dtfa v^o ba't dang thtfc, n e n ngiTdi ta thiTdng goi la

phi/cfng phap bat dang thiJc

2 T a c d the sOr dung ba't dang thifc Bunhiacopski de giai nhu" sau:

T h e o bat dang thi?c B u n h i a c o p s k i ta c d : 1 ;'

x l + V 4 - x ^ l l < [ x ^ + ( 4 - x ^ ) | ( l ^ + 1 ^ )

= > x + V 4 - x ^ <2sf2 ( 5 )

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Chuyen BDHSG Toan gia tr| Idn nha't g'A tr| nh6 nhat - Phan Huy KhJi

+ 1 + 1 > 3z

Tir do va diTa vao gia thie't x + y + z = 3 suy ra: + y ' + z^ > 3 (2)

Dau bang trong (2) xay ra o X = y = z = 1

2^x y + y z + z X j + (xy)(yz) + (xy)(zx) + (yz)(zx)

Theo bat d i n g thtfc Cosi, ta c6: i

x^y^ + y^z^ + z^x^ > (xy)(yz) + (xy)(zx) + (yz)(zx) (4)

z" + z'' + z > 3z^

3 ( x ^ y ^ + y ^ z ^ + z ^ x ^ ] Tir (2), (3), (4) suy ra: P > 4^- — — ( hay P > 1. (5)

3 ( x ^ y^ + y V + z \ ^ j

De thay dau b^ng trong (5) xay ra o X = y = z > 0

^ Vay min P=l <i>x = y = z > 0

Bai 15 Cho x, y, z la cac so thifc diftftig T i m gia tri nho nhat ciaa bieu thuTc

+ 8 y z >/y^ + 8zx + ^xy

Hiidng ddn giai

V i e t lai P difdi dang sau:

Cty TNHH MTV DWH Khang Vi§t

P x-y/x^ + 8 y z y-y/y^ +8zx z^z^ + 8 x y TO'(1) va theo bat dang thiJc Svac-xd, ta c6:

De thay da'u bang trong (5) xay r a o x = y = z = 1

Nhqn xet: Ta c6 bai toan IMng tU" sau:

Cho X > 0, y > 0, z > 0 va X + y + z = 1 T i m gia tri nho nhat cua bieu thtfc

Is!, Jr ;

I P = - + •

x ' ' + 8 y z y-^+8zx z ^ + 8 x y

X' +8xyz y +8xyz /: +8xyz

A p dung ba't dang thuTc Svac-xd, ta c6: P > (x + y + z)

x' + y + z" + 24xyz Theo bai tren ta c6: (x + y + z ) ' > x V y ' + z' + 24xyz

ChuySn BDHSG Toan gia tri I6n nha't va g\& tri nh6 nha't - Phan Huy KhSi

Dau bkng Irong (5) xay ra <=> x = V 4 - x ^ o x = V2

Vay maxy = y/l C:>x = yl2

Cdch 2: (PhiTOng phap chieu bie'n thien ham so)

Xet ham so f(x) = x + V4 - x^ vdi - 2 < x < 2

Do 4 - 2x' > 0 khi 0 < X < yfl va 4 - 2x' < 0 khi N/2 < x < 2, nen ta c6 bang

bien thien sau:

0 ^/2 ^ ^ ^ ' 2

T i r d o s u y r a max l"(x) = I(N/2) = 2N/2 ;

• " ' ' ^ " min l ( x ) = min{l'(-2);r(2)) = min(-2;2) = - 2 i > :

- 2 < ,\ 2

Nhgn xet: Ten goi cua phifdng phap hoaii toan phan linh di'ing qua each giiii vifa

trinh bay c'J Iren

Cat7i J; (PhifcJng phap UMng giiic hoa) i

Xet ham so t(\) = \+ ^ - x ' \(U - 2 < x < 2

Do - 2 < X < 2, nen dat x = 2sin(p vdi - - ^ < cp < ^

TCr do ta quy ve xet ham so

F((p) = 2sin(p + >/4(l - sin" (p) 2sin(p + 7 4 c o s ' cp = 2sin(p + 2|eos(p

= 2sin(p + 2cos(p (do khi - - ^ < cp < ^ ihi coscp > 0)

Cdch 4: (Phu'dng phtip mien gia trj ham so)

Gia sur m hi mot gia trj tiiy y cua ham so \'(x) = x + \l4-x' vdi - 2 < x < 2

Khi do phu'dng trinh an x sau day x + \J4-x~ = m (1) c6 nghiem

Ro rang (1) o \ / 4 - x - = m - x (2)

B ^ i loan ltd thanh: Tim m de (2) c6 nghiem ^ " '

(2) CO nghiem khi vii chi khi difclng cong y = SJA-X^ va diTcJng thang y = m - x

cat nhau n : , v - , ! - - x ,

De lha'y y = m - x o x + y = m, con y = ^ 4 - x " c^< ^ ~^

I y > 0 Vay ta can tim m de du'c'Ing thang x + y = m va nifa du^clng Iron x^ + y" = 4 (phan nam phia tren Iriic hoiinh cat nhau)

Dc tha'y dieii nay xay ra

khi va chi khi du'cVng

thring X + y = m nam giiJa hai du'clng x + y = - 2

Chuygn dg BDHSG Toan gia trj Icin nha't gia tri nh6 nhat - Phan Huy Khtii

Nhdn xet: Cach giai tren diTa vao each tim gia tri cua ham so day c6 ket hdp

them phifdng phap suT dung do thi va hinh hoc), vi the ta c6 the noi rang da

sur dung phi/dng phap mien gia tri ham so' de giai bai toan tim gia trj \6n nhat

va nho nhat noi tren

Binh ludn: V d i bai loan 1, la da su* dung bon phu'cfng phap khac nhau de giai

bai loan tim gia trj Idn nhat va nho nhat cua ham so M o i phU'dng phap deu

CO nhffng ifu diem rieng cua no

Bai toan 2: Cho x > 0, y > 0 Tim gia tri Idn nhal va nho nhat cua bieu thiJc P =

( x - y ) ( l - x y ) •: ^ • • ' • • '

(l + x ) ^ l + y)2 •

(De thi tuyen sink Dai hoc, Cao ddn^ khoi D )

( i M I HuAng ddn giai

CacA 7; (Phufdng phap ba'l dang ihiJc)

lha'y P c6 the vie't lai dudi dang sau day

Do y > 0, nen lij" (1) suy ra P < - , V X > 0, y > 0 P = - x = 1; y = 0

TiTOng liTlai c6 ^' '< 'f' i5«»f^« 'flfffetv''}l>_'n^frflJ j s l i i i f U '

Do X > 0, nen liJT (2) suy ra P > - - V x > 0; y > 0 P = - - o x = 0; y = 1

Tom lai max P = — < = > x = l ; y = 0; minP = <=>x = ();y = 1

Cach 2: (PhiTdng phap bat dang ihu-c)

Ta c6: ( x - y ) ( l - x y )

( l + x ) ^ l + y)2

X - y 1 - xy (l + x)^(l + y)^

Mat khac d i lha'y [(x + y) + (1 + xy)]^ > 4(x + y)(l + x y ) (4) Dau bang trong (4) xay ra « x + y = 1 + xy

( x - y ) ( l - x y ) Tir(3) & (4) di den

4 4

Nhdn xet: Cung suT dung phiTcMg phap ba't dang thiifc, nhu-ng ta co 3 each giai khi

nhau bai loan tren

Cach 4: (Phifcfng phap lifting giac hoa) '1 i

ChuySn BDHSG To^n gi^ tr| Idn nha't vA gia trj nh6 nha't - Phan Huy Khii

K h i do P = , ^ ^ " ' f , = l a n ^ a c o s V - tan^pcos^p

(1 + tan^ar (1 + lan^ p)^ • , = s i n W o s ^ a - s i n ' P c o s ' P = - s i n ' 2 a - — sin'2p

liinh luan: V d i b i i i l o a n t r c n l a c 6 4 each g i a i k h a c n h a i i , I r o n g d o c 6 3 each

c u n g siir d u n g phiTdng p h a p ba'l d a n g ihiJc ( b a e a c h n a y l a i k h t i c n h a u ) Q u a

do la t h a y r o l i n h d a d a n g c u a phiTdng p h a p d u n g d e t i m g i a t r i lofn nha't v a

n h o nha't eiia h a m so , , „,,

IJai t o a n 3: G i a silf X, y la hai so ihifc sao cho X" + y^ = 1

T i m g i a t r i UKn nha't va nho nha't ciia b i c u thiJc P = + 6 x y ) ^

1 + 2 x y + 2 y "

' (' , (De thi tuyen sink Dai hoc Coo ddn)> khoi B)

Hitdng ddn gidi ':dch 1: (Phifdng p h a p m i e n g i a t r i h a m so) ' i < { i

Chuy§n dg B D H S G Tpan gi^ tr| Ifln nhat gia tri nh6 nhat - Phan Huy KhSi

* Ncu y ^ 0, thi P = ^^^^ vdi t = - 6

t e E l e M U'W 'n'.l-.U:^:'' 'f.f • '

Vay maxP = 3, minP = - 6 khi x^ + y' = 1

Binh luqn: Tinh da dang cua cac phU'dng phap giai bai loan lim gia tri Idn nhat

va nho nhat cung the hicn ro qua thi du nay

Bai toan 4: Cho x > y, x > z va x, y, z e [ 1 ; 4].Tim gia tri nho nhat ciia bieu

LtJi giiii cua bai toan nay la su" kel hctp khco Ico cua hai phifcfng phap bat

dang ihiJc va chieu bien thien ham so nhif sau:

Viet lai bieu Ihtfc P diTdi dang: P = 1 1 1 • + +

Dau bang trong (2) xay ra khi va chi khi a = b hoac ab = 1

Do a > 0, b > 0, ab > 1, vay (3) dung suy ra (2) dung

(l + aKl + x/ab) (l + b)(i + 7^)

V I t > 1 ^ f (t) < 0 V t e [ 1 ; 2] TO do c6 bang bien thien sau:

Chuygn dg BDHSG Toan gii trj I6n nha't va glA tr| nh6 nha't - Phan Huy KhAi

Do X, y, z e [ 1; 4] ncn P = — <=>x = 4, y = l , z = 2

Nhu" the minP = 34 33 ; ~" <=>x = 4;y = l;z = 2 33

Bai toan 5: Cho bon so ihifc a, b, c, d thoa man dieu kien a^ + b" = c' + d"^ = 5

Tim gia trj Idn nha'l cua bieu thiJc

P= >y5-a-2b + V 5 - c - 2 d +N/5 -ac - bd

Hii(fng dan gidi

Ldi giai hay nhat va dac sSc nhat cho bai loan nay la phu^dng phap su* dung

hinh hoc sau day:

Ta thay cac diem M(a; b), N(c; d) va Q( 1; 2) trong do a, b, c, d la cac so thifc

thoa man dieu kien dau bai deu nam tren difdng Iron c6 tam tai go'c toa do

va ban kinh bhng v5 ' f i"i: V ^ ^

Viet lai bieu iMc P dxidi dang sau: f ?

x^vlfi^m-P = /(a-l)2+(b-2)^ ka-cf+ih-df

(MQ + NQ + MN) =

d day CMNQ la chu vi cua tam giac MNQ

Ta sur dung ke't qua quen bict trong hinh

hoc phiing sau day: Trong cac tam giac

npi tiep trong mot di/dng tron ban kinh

R cho trU'dc, thi tam giac deu la tam

giac CO chu vi Idn nhat Mat khac tam

giac deu noi tiep trong du'dng tron c6

ban kinh R c6 canh b^ng R ^/3

VivayCMNQ< 3N /l5.Tir(l)suyra P < ^ ^ = - N / 3 0

v2 2

Do d6 maxP = 3^30

Qua 5 bai toan tren, chung toi da gidi thieu vdi c&c ban cac phiTdng phap?

chinh de giai bai toan tim gia tri Idn nhat va nho nhat cua ham so"

- PhiTdng phap bat d^ng thtfc

- PhiTcfng phap chieu bie'n Ihien ham so

- Phiftlng phap mien gia tri ham so 'i ^ j^ifif'^f^'M^^'•''•>::^r^' '

- Phi/dng phap lU'dng giac hoa c.^i

'.-rl Phi/ctng phap hinh hoc hoa ,;:^^^iry, h^> rti '.-rl

Cac ban cung da tha'y dtfdc chiing ta c6 the c6 nhieu phU'dng phap khac nhau de giai cung mot bai toan tim gia tri Idn nhat va nho nhat cua ham

§2 N H I N LAI CAC BAI T O A N V E GIA TRj L 6 N N H A T V A

Bai 1: (De thi tuyen sinh Dai hoc Cao dunf- khoi A-2011)

Cho X, y, z la cac so thiTc sao cho x > y, x > z va x, y, z e [1; 4) Tim gia trj

^ t X y z nho nha't cua bieu thuTc: P = + + 7 "• :: -

Tir do ke't hdp hai qua trinh tren ta se di den IcJi giai cho bai toan Van d6

la d cho viec phat hien ra (1) khong phai la dieu de dang

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Chuy6n dg BDHSG Toan gii trj I6n nha't va gi^ tri nh6 nha't - Phan Huy Kh5i

2 Thay cho vice suT dung mot bat dang thtfe phu, ta co each lam sau day c6

ve "tiT nhien " hcJn mot chiit

Coi P nhiT la mot ham so'eua z, xet ham so'an z

* Ne'u X > y (chii y la x > y, nen khi x y thi x > y) thi x - y > 0 nen

ii^.'"' P'(z) = 0<=>z^-xy = 0<=>z= ^xy • Hi!K a i a j ' , ; ; : ; J J : - : ^ ' ^

Ta CO bang bien thien sau (suy ra tif (1)) ? , ! ' ! 7 z

5 33

34 nen minP = — 33

Ro rang viec phat hien ra (2) theo cdch giai nay "tif nhien" hdn trong

each giai cua bai 4, mat du no phuTc tap hcfn ve mat tinh toan!

Cty TNHH MTV DWH Khang Vigt

Bai 2: (De thi tuyen sink Dai hoc Cao ddrtfi khoi 8-2011)

Cho a, b la hai so thiTc dUdng thoa man dieu kien:

2(a^ + b^) + ab = (a + b)(ab + 2) Tim gia tri nho nha't cua bieu thtfc P = 4 + 9 b^^

+

b^ a^

Difa P ve dang sau: P = 4

Hiidng dan giai

fa h\ — + —

Vb &)

= 4 — + — a b

.b a j Viet lai gia thie't diTdi dang sau: 2

- 3 + 9

Ta c6: f'(t) = 12t^ + 18t -12 = 6(2t^ + 3t - 2) va c6 bang xet dau sau:

Chuyfin dg BDHSG Join g'lA tr| Ifln nha't vt g\i trj nh6 nha't - Phan Huy KhAi

Da'u bkng irong (7) xay ra khi va chi khi

Bang each kc'l hdp khco leo giffa dieii kien va bat dang thiife Cosi la suy ra

dicu k i c n (6) Con lai d l nhicn la silr dung phu'dng phap ehicu bien thicn ham

so" ham so dc giai bai loan

B a i 3: Cho eac so' thi/e khong am a, b, c thoa man a + b + e = 1

T i m gia I n nho nha't cua bicu ihtfe: /

M = 3 (a'b^ + b'e' + e'a^) + 3 (ab + be + ca) + 2 Va^Tb^TiJ

(De thi tuyen sinh Dili Iwc, Cat) dcinii; khoi B ~ 20J0)

HUcing ddn giai

Ro rang ta CO bat ddng thiJe hien nhicn sau t

Ta lai eo a^ + b ' + e' = (a + b + e)' - 2(ab + be + ea) = 1 - 2(ab + be + ca) (2)

T i r ( l ) ( 2 ) s u y r a

M > (ab + be + ea)^ + 3(ab + be + ca) + 2 ^ 1 - 2 ( a b + be + ea) (3)

Da'u bang Irong (3) xay ra khi va chi khi c6 dau bang trong (1), ttJc la khi va

chi khi ab = be = ca ,

Cty TNHH MTV DVVH Khang Vi^t

(diiu bang xay ra ehi lai t = 0), n c n r'(t) la ham nghich bien trcn |(); - J

Binh ludn: V i c e x c t dai lu'dng phu thu()e vao bien ab + be + ea la mot y nghia

hoan toan tiT nhicn D i c u do dan den vice x c t eac he thuTc (1) va (2) D e n

1 day vice xet ham so': r(t) = t " + 3t + 2 V l - 2 t vdti 0 < t < -

va suf dung phiTdng phap chieu bien thicn ham so de giai b a i toan la mot vice lam tiT n h i c n ! Cach giai vifa trinh bay ro rang la each giai l o i U\ nha't

Bai 4: (De thi tuyen sink Dai hoc, Cao clan,^ khoi D - 2010)

mien xac dinh eija no

Chuy§n dg BDHSG JoAn gia tri Idn nha'l g\& trj nh6 nhS't - Phan Huy Khii

Tur (2) suy ra > 2. Do y > 0, nen C O y > V2 , V X e [-2; 5] (3)

Da'u bang trong (3) xay ra o (a + 3)(5 - x) = (x + 2)(7 - x) <=> x = j

Tuf do suy ra miny = ^/2 cs> x =

Binh luan:

1 Viec ph^t hien ra y > 0 V x e [-2; 5] la le tif nhien va dcfn gian Tir do ta

stjr dung tinh chat cd ban sau nay:

N e u f(x) > 0 V x e D , thi min f ( x ) = / m i n f ^ ( x )

V i e c phat hien ra (2) la dieu binh thiTcJng v i chi dung eac phep bie'n doi

ddn gian ve can thiJc da hoc kT d cap 2

f \ PhU'dng phap bat dang thiJc ap dung d day tuy ddn gian nhU"ng c6 hieu qua

Idn Theo chiing toi do la each giai hi^u hieu nhat doi v d i bai loan nay

2 Ta hay so sdnh each giai tren vdi each giai khae sau day (suT dung phiTdng

phdp chieu bien thien ham so)

3 "

T r d n cd s6 66 ta c6 ngay T > 0 k h i - < x < 2, va

T = (4-2x )Vx^ + 3x +10 ( 3 2 x ) V x ^ + 4x + 21 k h i 2 < x <

-2 (2x - 3)V-x^ +4X + 21 - (2x - 4)V-x^+3x + 10 khi 2 < x < 5

Difa vao nhan x6t sau neu A > B > 0 thi A > B o >

Cty TNHH MTV DWH Khang Viet

Ta thu lai ke't qua tren

Ve tinh hieu qua cua hai phiTdng phap trong hai each giai xin danh phan binh luan cho ban doc (ve nguyen l i thi rat ddn gian chi can xet da'u y ' , nhU'ng

-BM 5: (De thi tuyen sink Caoddnf^khoi A-2010) i

Cho hai so' thU'c difdng x, y thoa man dieu kien 3x + y < 1

L a i theo ba't dang thtfc Cosi, ta co: yjlxix + y) < = ^— •

Tiif do thay vao (2) va c6: - + > — ^

(3)

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:huy6n 6i BDHSG ToAn g\i lr| Idn nhS't va gi^ tri nh6 nhS't - Phan Huy Kh4i

Da'u bang Irong (3) xay ra <=> 2x = x + y <=> x = y

3x + y Da'u bang Irong (4) xay ra o dong th(li c6 da'u bang trong {1), (2), (3) <=> x = y

T h c o gia ihic't i h i 0 < 3x + y < 1, nC-n lir (4) co A > X ( 5 )

Jai 6: (De thi tuyen s'mh Dai hoc Cao dcfni^ kiwi D)

Cho a i c so thifc x y khong am va thoa man x + y = 1 T i m gia tri Idn nha't va

nho nha't ciia b i c u thuTc S = (4x- + 3y)(4y- + 3x) + 25xy

HUiiitg dan giai

Ta CO S = 1 6 x ' y ' + 12(x'' + y ' ) + 34xy = I f i x ' y - + 12(x + y)(x" - xy + y") + 34xy

16

M ^ l f,^ 251 25 )• = max < 12; — ^ = —

liiiih liigii: Ro rang phufitng phap chicu bicn thicn ham so la thich hdp nhal dc

giiii bai loan nay

Bai 7: (Dc thi tuyen sinh Dai IJOC Cao dan}' khoi B)

Cho Ccic so ihiTc x, y lhay d d i thoa man d i c u k i c n (x + y ) ' + 4xy > 2

T i m gia Irj nho nha't ciia b i c u ihiJc A = 3(x"' + y ' + x ' y ' ) - 2(x- + y ' ) + 1

HUiing ddii giai

V i c t lai A d i / d i dang A = ^ ( x ' * + y ^ ) + | ( x ' ^ + y-*) + 3 x - y - - 2 ( x - + y " ) + l

Da'u bhng trong (3) xay ra o dau bang trong (2) xay ra o x ' = y <=>

Do (X + y ) - > 4xy, ncn lir gia ihict (x + y)' + 4xy > 2, suy ra f>^\

Chuyen dg BDHSG Join gia trj I6n nha't \ii g\A trj nh6 nha't - Phan Huy KhSi

Bi?ih ludn: V i e c diTa ve xet b i e u thtfc d o i v d i x^ + y^ la mot le tiT nhien

Bang phu'dng phap bat dang thtfc ta d i den danh gia (3)

Cong viec con l a i d i nhien can den viec van dung phiTdng phap chieu b i e n

thien h a m so'

Ta thay viec ke't hdp nhuan nhuyen giiJa hai phifdng phap bat d i n g thiJc va

chieu b i e n thien h a m so da dan den siT thanh cong trong qua trinh g i a i b a i

toan nay

Bai 8: (De thi tuyen sinh Dai hoc Cao ddn^ khoi B)

Cho X , y la cac so' thifc thoa man dieu kien x^ + y^ = 1 '

T i m gia t r i Idn nha't va nho nha't cua bieu thtfc P = ^ ^ ^ ^

l + 2 x y + 2y^

Hitdng dan giai

X e m I d i g i a i (ba each) trong bai toan 3, muc § 1 , chiTdng 1 cua sach nay

Binh luqn: B a i toan c6 den 3 I d i g i a i khac nhau (hai each suT dung phiTdng phap

m i e n gia t r i h a m so trong do c6 ke't hdp ca phiTdng phap li/dng giac hoa, m o t

each suf dung phu'dng phap chieu bien thien ham so') - i "

B a i 9: (De thi tuyen sink Dai hoc Cuo ddn}^ khdi D)

C h o x > 0 ; y > 0 , :

T i m gia t r i Idn nhat va nho nhat cua bieu thiJc P = ( x - y ) ( l - x y )

( l + x ) ' ( l + y ) '

Hudng dan giai

X e m Idi g i a i (bon each) trong bai toan 2, muc § 1 , chtfdng 1 cua cuon sach nay

Binh luqn: D a y la m o t trong cac bai toan the hien ro nha't tinh da dang cua viec

suT dung cac phu"dng phap khac nhau de g i a i mot b a i toan ve t i m gia t r i Idn

Cty TNHH MTV DWH Khang Vigt

nha't va nho nha't cua h a m so', l / u the cua phiTdng phap sijr dung bat d i n g thufc

ro net qua t h i du nay ' - ^ ' ;

(De thi tuyen sinh Caoddn}> khoi A) • ii-; ,v > * > ;

Cho x , y la hai so thi/c thoa m a n x^ + y^ = 2

T i m gia trj I d n nha't va nho nha't cua b i e u thiJc P = 2(x^ + y^) - 3 x y

Hudng dan giai

T a co: P = 2(x + y ) ( x ' + y ' - x y ) - 3xy = 2(x + y ) ( 2 - x y ) - 3xy (1)

T a c6: x ' + y^ = 2 (x + y)^ - 2xy = 2 x y = ^ ^ ^ ^ ^ ~ ^ • (2) Thay (2) vao (1) r o i dat t = x + y, ta c6:

.2

P = 2t 2 - t ^ - 2 ^

• A

_ 3 ! l _ l =-e-^t^ + 6t + 3 , , Bay g i d ta t i m m i e n xac dinh cua t: 7

Chuy6n 6i BDHSG loin g\ tri I6n nhS't va gia tri nh6 nha't - Phan Huy Kh^i

1 V i c e quy vc b i c u ihuTc d o i v d i x + y (ma la sc dai b a n g I) r o i stir d u n g

phifdng p h a p e h i c u b i c n i h i c n h a m so'dc g i a i la v i c e l a m u / n h i c n ( g i o n g

nhi/ irong nhicu bai loan x c l tri/c'tc day)

2 Ta ihur suy n g h l d i c u k i c n x~ + y" = 2 g(;i y la eo n c n siir d u n g phifttng p h a p

A p d u n g ci)ng Ihifc smtpcoscp = —

va dill I = sincp + eosip - V2 cosj <p - — (do do -\[2 < l < V2

la CO P = 4N/2I 1 - I - -1 - 6 r - 1

Khi do la h i i q u y vc biii l o a n l i m gia l i i ii'Jn n h a l va nho n h a l ciia m o l h a m

so lU"(tng ur n h i r d a l a m Irong phan I r c n Vay sir d u n g " l i r d n g g i a c hoa " c h i

: la p h a n d a u , c o n l i c p i h c o la iai si? d u n g phiriJng p h a p c l i i c u b i c n ihiC-n

h a m so n h i r d a l a m ('UrcMi

B a i I I : {De thi ttiycn Dai line (\ii> c/(?/;,^' klioi B)

Cho X, v, / la ba so dirt<ns: T m i uia Iri nho nhal ciia b i c u ihi'rc:

1 I ' - 1

X c l ham so 1(1) = + vi'fi l > 0 Ta ci) I'd) 1 —

-/••" ; f v - , 2 I I - r

( I ) (2)

Dau bang Irong (4) xay ra <=> x = y = z = 1 y; , ; ;, , , ; ' Vay minS = ^ ,^ : Gia irj nho nhal da I du'dc khi va chi khi x = y = / = I ''

1 M o l Ian nffa la lhay sir k c l hdp giffa phiTdng phap ba't d^ng ihiJe vii phuTilng phap chicu bicn ihicn ham so cho phcp la giiii bai knin ircn

2 Ta t o Ihc siir dung ihuan liiy phu'ctng phap ba'l dang ihii'c dc giiii bai loan

x ' I 3 nay T h i l l vay la sc chi^ng minh rang khi x > 0, Ihi — + — > — (5)

2 x 2

T h c i t vay (3) <=> x ' - 3x + 2 > 0 <=> (x - l ) ( x ' + x - 2) > 0

o (X - i r i x + 2) > 0 (6) R6 rang (6) diing Vx > 0 (5) dung V X > 0

Tir do suy ra S > - => minS = - < = > x = y = / = l

X c m ra each giai nay qua ddn giiin Nhu"ng irJi giiii ciia no khong ur nhicMi

& cho: lam sao ma b i c l difclc vc phiii ciia (5) la ^

^^y\2i {De thi tuyen sinh DiiUwc Cci() ddn}> khoi A)

Cho X, y la cac so ihirc khac 0 va ihoa man dicu kicn xy(x + y) = x ' - xy +

y"-T i m gia i r i Idn nhal ciia bicu ihiVe: A = + •

^ y

HUdiig dan giai : •

Do x 7^ 0 , y ; t 0 n c n xy(x + y) = x ' - xy + y"

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