0 +15 0 ^ 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
>BfensoantheonOidun^va
c^utrucd^tliicuaBOGDfiflT

• ^ V I E N Tl'NHBiNHT

O K ] zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

NHA XUAT BAN DAi HQC QUOC GiA HA NQI

Ha NQI

Download Ebook Tai: https://downloadsachmienphi.com
16 Hang Chuoi - Hai Ba Trang - Ha Npi
Dien t h o a i : Bien t a p - Che ban: (04) 39714896
Hanh chinh: (04) 39714899; Tong bien tap: (04) 39714897
Fax: (04)39714899

BaizyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
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

tim gid tri

Chiu

trdch

nhiem

i>Au

IJCU N6I

N H A X U A T B A N D A I H Q C Q U O C G I A H A NQI

nhiing cdu kho nhd't ciia de thi.

xuat ban

vi^c

nhd't, nhd nhd't ciia hdm .w thudng xuyen cd mgt vd thut'fng Id mot trong
,

, ., ,

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

doc • Tong bien tap : T S . P H A M T H I T R A M

Gidm

I

quan tdm ciia ban doc. Tnmg

cud'n sdchzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON
"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

vdi

hdi todn

Idn

vd

H A I NHtf

nhd't doi

Che ban:

Cong ty K H A N G V I E T

cdch dp dung hdi todn nay de gidi nhieu hdi todn lien quan den no.

Trinh

C o n g ty K H A N G V I E T

Bien

tap vd sita bdi:
bay bia :

Noi

trdch

nhi^m

ngi dung

vd ban

quyen

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

tim gid tri

nhd't

dung ciia cud'n .sdch dUOc trinh hdy trong

Chiiong 1

Chiu

nhiing

v(H

so" se gidi thi^u

tieu de

vdi

" Vdi

nhd nhd't ciia hdm so.cdng nhu hiet

chUcfng.

bdi todn md ddu ve gid tri l^n nhd't va nhd nhd't cua ham

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

Tong phdt

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

hdnh:

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

C6NG T Y TNHH MTV
Sm

ajP

cdc ban nhiing kien thiic chud'n hi can hiet di' doc tiep cdc chUifng sau ciia cud'n sdch.

Cdc

D ! C H vy V A N H 6 A K H A N G V I | T

nhd nhdt ciia hdm sd'duac trinh hdy tit chUOng
/^Dia chJ: 71 Dinh T i § n Hoang - P D a Kao - Q.1 - TP.HCM
~ ^
Dien thoai: 08. 39115694 - 39105797 - 39111969 - 39111968
Fax: 08. 3911 0880
Email: lWebsite: www.nhasachkhangvlet.vn
^

Idn
., •

phUcfng phdp ca ban vd thong dung nhd't de gidi bdi todn tim gid tri

Chitang 2: Phi/mg

2 den

chuang

6.

nhd't

vd

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.

SACH LIEN KET
CHUYEN DE BOI

ChiMng 5: Phiicfng phdp mien gid tri hdm
D J O N G

HQC SINH

G161

GIA TRI LdN NHAT,

tim gid tri Idn nhdt vd nhd nhdt cua

hdmsd.
ChUOng 6: PhUmg phdp

GIA TRj NHO NHAT.

sd

dS

thi vd hinh hgc tim gid tri Idn nhdt vd nhd nhdt cua

hdm sd,

Ma so : 1 L-31 7DH2012.
d mSi chuang, chung toi cdgdng truyen tai den ban doc n^i dung co ban cua phuc/ng
So lugfng in 2000 Wn, kho 16x24 cm.
phdp, dUa ra cdc Idp hdi todn md phuc/ng phdp gidi no la thich h(fp nhd't. Thdng qua vifc
In tai Cty TNHH MTV in an MAI THjNH DL/C.
Phdn tich, hinh luqn vd dUa ra lam doi chiing nhieu phUtfng phdp khdc nhau gidi cUng
Dja chl: 71 Kha Van Can, P.Hiep Binh Chanh, Q.Thu Dufc, Tp.HCM.
mQt bdi todn se giup cdc ban tim duoc cho minh mQt phuang phdp m vi$t nhdt de gidi
So xuat bin: 1297-2012/CXB/08-213/DHQGHN, ngay 26 thang 10 nam 2012 hdi todn gdp phdi. Do Id dieu mdi me cua cud'n .sdch ndy. Chung toi ludn ludn gia tinh
Quyet djnh xuat b i n so: 311 LK-TN/QD-NXBDHQGHN
thdn chii dao d'y trong tvCng phdn ciia cud'n .sdch.
in xong va n6p liAi chieu qui I nam 2013.

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ChMng

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

MdDltUVfGliHllllhllllllt

trong vi^c hi$n ludn phurdng day cdng Id mot chii de thi/dtng xuyen xud't hi^n trong cdc de thi tuyen sink vao Dai

VANHiNHltCUAHAnSdr

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

§ 1 . VAIBAITOANMdDAU

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

T r o n g m u c n a y c h u n g toi gidi thieu v a i bai toan v e gia tri Idn nha't va nho nha't

tham so.

Cudn sdch nay chu yeu trinh hay cdc hai todn tim gid tri
.so

Cty TWHH MTV D W H Khang Vi^

ciia h a m so. T h o n g qua nhffng hai toan nay, c h u n g toi muon d e c a p d e n c a c

nhat, nhd nhcft trong Dai

phtfdng p h a p c d ban nhat d e giai c a c bai l o a n v e gia trj Idn nhat v a nho nhat se

vd Gidi tich.

di/dc trmh b a y k y y o n g c u o n s a c h nay.

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

B a i t o a n 1: (De thi tuyen sink Dgi hoc, Cao dang khdi B)

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

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'

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

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

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

Hildiig dan giai

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

Cflc/i 7 ; (PhU'dng p h a p bat d a n g thtfc)

dieu hd ich khi doc no.

!

'

s

H a m so' d a c h o x a c d i n h k h i -2 < x < 2.

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

Tacd

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

x > - 2 ; V 4 - x ^ >0 V x

D o do f ( x ) > - 2 , V x e

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

L a i c6

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

f(-2) =-2

G

j>

[-2;2] ,

[-2; 2]
'^'fn -

;

(1)
; i,

H;

:J

nm)

M

(2)

T i l f ( l ) ( 2 ) s u y r a m i n f ( x ) = -2.

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

T a se chu-ng m i n h f(x) < 2V2

PHANHUYKHAI,
V i p n T o a n hoc, 18 DiTcfng Hoang Quoc V i ^ t - Quan C a u G i a y - H a Noi.

That vay (3) o

X i n chan thanh c a m dn.

V x e [-2; 2]

X + V 4 - x ^ < lyfl

o

(3)

V4-x^ < 2 > ^ - x ' — -

c ^ 4 - x ^ < (2V2-x)^ ( d o x < 2 ) o 2 x ' - 4>/2 x + 4 > 0

Tacgia

«

(X -

72 )^ > 0.zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ
'
(4)

Tur ( 4 ) suy ra ( 3 ) dung. Nhu" v a y ta c 6 f(x) < 2^/2
L a i cd
Nhdn

f(V2) = 2N/2 , n e n

max f(x) =
-2
xet:

V x e [-2;

2].

2V2 .

,

1. C a c h g i a i tren h o a n toan dtfa v ^ o ba't d a n g thtfc, n e n ngiTdi ta thiTdng g o i la
I,'

phi/cfng phap bat d a n g thiJc.

,

i

2. T a c d the sOr d u n g ba't dang thifc B u n h i a c o p s k i d e g i a i nhu" s a u :
T h e o bat d a n g thi?c B u n h i a c o p s k i ta c d :

x.l + V 4 - x ^ l l

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

= > x + V 4 - x ^ <2sf2.

Tron Bo SGK: https://bookgiaokhoa.com

1 ;'

(5)

Chuyen

BDHSG Toan gia tr| Idn nha't

Download Ebook Tai: https://downloadsachmienphi.com

+ 1 + 1 > 3z.
Tir do va diTa vao gia thie't x + y + z = 3 suy ra:

+ y ' + z^ > 3.

P-

(2)

x-y/x^ + 8 y z

D a u bang trong (2) xay ra o X = y = z = 1.

y-y/y^ + 8 z x

(I)

z^z^ + 8 x y

TO'(1) va theo bat dang thiJc Svac-xd, ta c6:

T i i r ( l ) , ( 2 ) s u y r a P > 1.

(3)

D a u bang trong (3) xay ra <=> dong thdi c6 dau bang trong (1), (2)

(x + y + z ) '

'
x^jy}

<=>X = y = Z = l .

+ 8yz + y ^ y ^ + 8zx + z-y/z^ + 8xy

,

.

(2)

-.

Z^.,-,

A p dung bat dang thiJc Bunhiacopski, ta c6:

V a y m i n P = 1 <=>x = y = z = l .
B a i 14, Cho x, y, z la cac so' thifc di/dng. T i m gia tri nho nhat ciaa bieu thtfc
y^

r.
2

Cty TNHH MTV DWH Khang Vi§t

g'A tr| nh6 nhat - Phan Huy KhJi

z^

2 2

y +yz + z

X/X.N/X^X^ + 8 y z + ^/y.^/y^/y^ +8zx + N/Z.VZI/Z^ + 8 x y

2

2

z +ZX + X

2

(x + y + z) x ( x ' + 8 y z ) + y ( y ^ + 8 z x ) + z ( z ^ + 8 x y )

x +xy + y
Hiidngddngiai

;;':..(•'•...!;

'-„4/.!'

j,,

= (x + y + z ) ^ x " ' + y ^ + z ' ' + 2 4 x y z j .

.

A p dung ba't dang thiJc Cosi, ta c6:

*

'

'

(3)

,, .

(4)

V i e t l a i P dtfdi dang:
4

4

^1

P=

+

x ^ ^ y ^ + y z + z^j

4

y_
y^^z^+zx + x ^ j

.

I

+

(1)

,

> x ' + y-'+ z ' + 277xyz.\/xVz^ - 3xyz
hay (X + y + z ) ' > x ' +

*

2^x

(2)

i

(3)

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

(4)

z" + z'' + z > 3z^
3(x^y^+y^z^+z^x^]
—
— ( hay P > 1.
3(x^y^+yV+z\^j

4^-

Nhqn xet: Ta c6 bai toan IMng

(5)

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

I

P =

Hiidng ddn giai
V i e t l a i P difdi dang sau:

y-^+8zx

Is!,. Jr. ;

tU" sau:

z^+8xy

Ta giai nhu" sau: P = —
+
X' + 8 x y z
y +8xyz
A p dung ba't dang thuTc Svac-xd, ta c6: P >

Tilf (**), (***) suy ra: P >

+ ^xy

(5)

-+ •

x''+8yz

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

>/y^ + 8zx

^^ ^
= 1.
(x + y + z)^

z^
/: + 8 x y z
(x + y + z)
x' + y + z" + 24xyz

(*)

(**)

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

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

+8yz

,

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

x' + y' + z ' > x y + y V + z V

Tir (2), (3), (4) suy ra: P >

, .

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

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

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

+ z' + 24xyz.

Thay (3), (4) vao (2) va c6: P >

x^+y^+z^+2(xy+yV+zV)
hayP>

• ,

(X + y + z)^ = x^ + y ' + z' + 3(x + y + /)(xy + y/ + /x) - 3xyz

z^(x^ + xy + y ^ j

A p dung bat dang thufc Svac-xd, ta c6:
fx^+y^+z^f
P>
i
L
x^(y^ + y z + z^) + y^ (z^ + z x + x ^ j + z^(x^ + x y + y^)

•

(x + y + z)^

" \y P >

(x + y + z)-

1=

1.

x +y + z

Vay min P = 1 o X = y = z = ^ .
^ a i 16. Gia siif x, y, z la ba canh cua mot tam giac c6 chu v i bang 12.

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1 ^ I

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ChuySn

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

Cty TNHH MTV DVVH Khang Vijt

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

- 2 < F((p) < 2V2 V - | < ( p < ^ ,
Vay maxy =zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
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
F((p) = - 2 <=> cos

v4-x^-x

X

Ta CO l"'(x) = 1

I I S

/4-x^

71

F((p) = 2%/^ <=> cos

= 10

2

\/4-x'

max Hx)=

'5

max

-2
= — <=> X = N/2

4

V2

71
CP--

i

Vay

9

71

<=> (p — =
^ 4 4

,

;

37t

•
71

o

(p = — <=>x = - 2 .
'

2

F((p) = 2>/2;
Tt

7t

R6 rang k h i - 2 < X < 0, thi f'(x)> 0.
Xet khi 0 <

X

< 2, ta c6 (4 - x") - x" = 4 - 2x-.

min

• -,

f(x)=

min
~
2

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

F((p) = - 2 .

<(n<
2

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

bien thien sau:
0

^/2

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

^^^'..2.

Khi do phu'dng trinh an x sau day x + \J4-x~
Ro rang (1) o

=m

(1) c6 nghiem.

\ / 4 - x - = m - x. (2)

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

max l"(x) = I(N/2) = 2N/2 ;

"

min l(x) = min{l'(-2);r(2)) = min(-2;2) = - 2

i

>

:

vdi - 2 < x < 2.

^"

'

(2) CO nghiem khi vii chi khi difclng cong y = SJA-X^ va diTcJng thang y = m - x
n . . . : , . v - , ! - . - x ,
cat nhau.

- 2 < ,\ 2

Nhgn xet: Ten goi cua phifdng phap hoaii toan phan linh di'ing qua each giiii vifa
trinh bay cJ' Iren.

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

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 < ^ .

F((p) = 2sin(p + >/4(l - sin" (p)

(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

TCr do ta quy ve xet ham so
2sin(p + 7 4 c o s ' cp = 2sin(p + 2|eos(p

giiJa hai du'clng x + y = - 2
va X + y = 2 V2 , ti'fc la

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

khi vii chi khi - 2 < m <
2V2 .(3)

= 2N/2COS((P--).
,^

n

n

~^

Tir(3)suyra

37t

71

rt

Do — < ( p < - => - • — < ( p - - < - .
2
2
4
4 4
Tif do suy ra - — ^ < cos

f(P--

;,i;.f-

max r(x) = 2%/2;
-2
min
<1

f(x) = - 2 .

-2
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Chuygn dg BDHSG Toan gia trj Icin nha't

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: Vdi 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

T i i r ( l ) ( 2 ) suy ra

(x-y)(l-xy)
(l + x ) ' ( l + y)^

•: ^

•• ' •

•

'

+ y)(l + xy)

(3)

•

(4)

Dau bang trong (4) xay ra « x + y = 1 + xy.
(x-y)(l-xy)
Tir(3) & (4) di den

Bai toan 2: Cho x > 0, y > 0. Tim gia tri Idn nhal va nho nhat cua bieu thiJc P =
(x-y)(l-xy)

(X

i2
[(x + y) + (l + xy)]

Mat khac d i lha'y [(x + y) + (1 + xy)]^ > 4(x + y)(l + x y ) .

CO nhffng ifu diem rieng cua no.

(l + x ) ^ l + y)2

Cty TNHH MTV DWH Khang Viet

gia tri nh6 nhat - Phan Huy Khtii

Lai

CO

P

•

4

(l + x ) 2 ( l + y)^

4

hay - i < P < | .
4
4

(5)

x = l ; y = 0, khi d6P = -

xy = 0

[xy = ()

x + y = l + xy

[x + y = l

x = 0;y = l , k h i d6P = - i

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

. i

M

Tom lai maxP = - < = > x = I ; y = 0; minP = — <=> x = 0; y = 1.
4
4

HuAng ddn giai

I

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

CacA J ; (Phu'dng phap ba'l dang Ihtfc)

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

y

(1 + x)^

(1 + y)^

P =

_

X

1

(1 + x)^

4

y

AB .

1

ta co: ( x - y ) ( l - x y ) ^ ( x - y . 1 - xy)
2/1 , . , \
n j_ v^2/-l . v"!^
(l + x)"(l
+ y ) ' A4(l
+ x ) ' ( l + y)

(1 + y)^ ^ 4

( x - y + l - x y ) 2 ^ (l + x ) 2 ( l - y ) '

4 x - ( l + x)'
4(1+ x)'

1 - i _ (x-ir
4 ~ 4

2

(1 + y)

4(l + x)2

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

X>

TiTOng liTlai c6

'f'

^'

'<

1

P=
(1 + x)

1

,

y

(1)

{\ yf

0, y > 0. P = -

x = 1; y = 0.

i5«»f^« 'flfffetv''}l>_'n^frflJ j s l i i i . f U '

X

^ (y-1)'

(1 + y)^ " 4

(2)

4

Lai eo

• '

4(l + x ) ^ l + y)-

_ 1

4(l + x)2(l + y)^

4

Mat khac P - - - o x = ( ) ; y = 1. V a y n e n P =
4

(l + x ) ^ l + y)2

(l + x)^(l + y)^

Dafu bang trong (2) xay ra o xy = 0.

1
4

o x = 0 ; y = 1.

nhau bai loan tren.
(1) ,

Cach 4: (Phifcfng phap lifting giac hoa)
Ta co: P =

Do X > 0; y > 0, ncn hien nhien la c6
x - y | | l - x y | < ( x + y ) ( l + xy)

4

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

Cach 2: (PhiTdng phap bat dang ihu-c)
- y 1 - xy

(doy>0)

. p _ , U - y K l - x y ) > _ ivx>(),y>0.
(l + x)^I + y)^

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

X

<4

Do vai iro binh dang giffa x va y, nen la co
(l + y ) 2 ( l + x)'

(x-y)(l-xy)

(i + y r

Mat khac P = - <=>' ^ ^ . VaymaxP= - <=>x = l ; y = 0.
x=l
4
4

(y-x)(l-yx) ^ 1

Ta c6:

4y

1--

Tird6suyraP^^^-y^/^-^^[4 Vx.O;y^O
(l + x)^(l + y ) ' 4

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

:

(2). .i

'1

i

(xem each 1).
(1 + x)^

(1 + y)^

Do X > 0; y > 0, nen dat x = tan^a, y = tan^(3, ( ) < a < - ; 0 < p < - -

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ChuySn

K h i do P =

,

(1 + tan^ar
,

^ ^ " ' f , = lan^acosV -

(1 + lan^ p)^

- - < P < 4
4

L a i l h a y P = ^<=><

P=

—
4

«

sau d a y ( a n t )

— sin'2p.

sin2p = ()'^

p =o

- 1

*

y = ()'
*

a =0

x-O

sin2p=I

m i n P = - - ^ x = 0 ; y = 1 .

i*>

d o 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 c h o X " + y^ = 1.

':dch
Do

- 6 < m < 3.
,/ /

D o m l a g i a t r j t u y y c i i a r(t), n e n t i i " ( 4 ) suy r a

,r

Ket hdp

.i

< {i

.

•'

V d i d i e u k i e n x ' + y^ = 1 i h i m a x P = 3, m i n P = - 6 .
Cat7i 2 ; (PhU'dng p h a p m i e n g i i i t r i h a m so)

2sin" a + 12sintteosa
1-cos2a+ 6sin2a
— =
^ 1 + 2sinacosa +2cos'a
sm2a + cos2a + 2 .

o

, ,
1-cos2a + 6sin2a
—=m
sin 2 a + cos 2 a + 2

;

( 6 - m ) s i n 2 a - (1 + m ) c o s 2 a = 2 m - 1.
-> •.,

—

/ \

-2V3

t"+2t +3

(4)

T i r d o suy ra m a x P = 3, m i n P = - 6 k h i x - + y~ = 1.
,

, (1

;

X

.

day I = — va t
y

e

(l)

Cdch

3: (PhiTctng p h a p c h i e u b i e n ihiC-n h a m s o )

TacoP^

2(x^.6xy)^
X

*

,

(3)

m ) " + (1 + m ) ' > ( 2 m - 1)"

c ^ 2 m - - 3 m - 9 < ( ) o - 6 < m < 3 .
2t- + i2t

,

1 - cos2a + 6sin2a = m(sin2a + cos2a + 2)

(3) CO n g h i e m 0 ( 6 -

+ 6

..^

C O n g h i t M i i . D o |sin2a + c o s 2 a | < V2 , V a e |(), 2TC|

Tir do (2)o

1. N e u y = 0 ( k h i d o x = 1). L u c n a y P = 2.

X

'

D o X ' + y " = 1, n e n l a d a l x = s i n a . y = c o s a , v d i a G |(); 271].

X e t h a i k h a n a n g sau:

X

^

=> s i n 2 a + c o s 2 a + 2 > 0 V a 6 |(); 27i|.

(1)

x^ + 2 x y + 3y^ '

'

P = 2 k h i y = 0, l a d i d e n k c l l u a n :

K h i d o phu'dng t r i n h sau d a y ( a n a )
'

(4)

, 1 !

G o i m l a g i a t r i t u y y c i i a P.

ddn gidi

2(x^+6xy)

+ y " = 1, n e n ta e o : P =

2. N c u y ^ 0. K h i d o P =

k h i d o (3) c 6 n g h i e m k h i va c h i k h i A ' > 0

V i i y ( 3 ) CO n g h i c m k h i v a e h i k h i - 6 < m < 3.

, .
KhidoP=

+6xy)^
1 + 2xy + 2y"

1: ( P h i f d n g p h a p m i e n g i a t r i h a m so)
X'

N e u m^l,

(De thi tuyen sink Dai hoc. Coo ddn)> khoi B)
Hitdng

(3)

m a x P = max r(t) = 3 v a m i n P = m i n r(t) = - 6 .
y*()
ItR
y*l)
lelR

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

T i m g i a t r i UKn nha't v a n h o nha't c i i a b i c u thiJc P =

:

N e u m = 2 , k h i d o 2 ( m - 6 ) ^ 0, nen ( 3 ) c 6 n g h i c m . V a y m = 2 la m o t g i a

c:> m ' + 3 m - 18 < 0 o

y = i

,

• » 2 1 ' + 12l = m ( t ' + 2t + 3 )

t r i c u a h a m so r(t).

V d i biii loan t r c n la c 6 4 each g i a i khac nhaii, Irong d o c 6 3 each

' ('

( 2 ) C O n g h i c m . D e tha'y v i t ' + 2 l + 3 > 0 V l ,

<=>(m - 2 ) t ' + 2 ( m - 6)1 + 3 m = 0 .
X

4 <=> i

sin 2 a = 0

:'

(1)

nen (2)

n

V a y m a x P = ^ < z > x = l ; y = ();
luan:

.

-•^'^ "^'^^ = m
t^+2t + 3

a =—

k h i d o phiTcIng t r i n h

t^+2l4-3

Va, pG|();-).
2

sin 2a = 1

-li—Ili^,

G o i m la g i a t r i l u y y c i i a h a m so 1(1) =

tan^pcos^p

•

= sinWos^a-sin'Pcos'P=-sin'2a-

TOr(l)suyra

liinh

Cty TNHH MTV DWH Khang Vi^t

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

j

(xemcachl).

+ 2xy + 3y"

N e u y = 0, l h i P = 2.

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"

;

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Chuy§n dg BDHSG Tpan gi^ tr| Ifln nhat

* Ncu y ^ 0, thi P =

^^^^ vdi t = y

Ap dvng (2) vdi a = ^ ; b = f.

6

r +21 + 3

Dat f(t) =

t G R thi f'(t) t^+2l + 3

-81^ + 121 + 36
,
( l ^ + 2 t + 3)^

, 2t^-3t-9
4
( t ^ + 2 t + 3)^ '

1+^
V

2

-00

ii

iH:

3

Kb. do a > 0, b > 0 va ab = - . 1 (do x . y),
(4)

I_+-L->-^

nen ta c6:

Ta CO bang bie'n thien sau:
I

Cty TNHH MTV D W H Khang Vigt

gia tri nh6 nhat - Phan Huy KhSi

l ^ ' ^ 1+
z
z _

+O0

^
Dau bang trong (4) xay ra

0

I'd)
1(0

+

2
2

Tif (4) ta

CO

1

P>

U'W 'n'.l-.U:^:''

leM

'f.f

1+

X

• '

=z

x=y

2 y y

+

2+3^

Tir do suy ra max f(l) = 3 va min t'(t) - - 6 .
teE

y

-

0

X

(5)

X

Vy

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

7y^ = z

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

Dau bang trong (5)xayra

x =y

A

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
thiJc: P =

x

y
y+z

2x + 3y

z
z+X

(De thi tuyen sink Dai hoc Cao ddiifi khoi A - 2011)

i

....

i

;l

I

Datt = E . D o x > y v a x . y e l l ; 4 ] n e n s u y r a l < ^ < 4 = ^ l < t < 2 . K h i d 6
\
t
2
?>—!— +—-hay P > - ^
+ 7~T3
1+t
21^+3 l + l
2+

Hii(fng dan gidi
LtJi giiii cua bai toan nay la su" kel hctp khco Ico cua hai phifcfng phap bat

Xet ham so' f(t) =
21+3
61

dang ihiJc va chieu bien thien ham so nhif sau:
Viet lai bieu

Ihtfc

1

P diTdi dang: P =

1
•+

2 + 3^
•

' ' '

'

•

X

1

(1)

+

1+^
y

Taco: f'(t) = ^

1 + '^

VI t > 1 ^

Tru"(1c hel ta chiJng minh bat dang thiJc sau: *
" ' *'
1
1
2
Neu a > 0, b > 0 va ab > 1, Ihi la c6:
+
>
1 + a 1 + b 1 + x/ab

f'(t)
f(t)

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

Thatvay(2)
> 0 o

<=>

1+a

l + >/abJ

1^1 + b

1 + Vab

>yab-b

Tab
(l + aKl + x/ab)

(V^-^/b)^^/^-l)
>0.
(l + a)(l + b)(l + >/ab)
Do a > 0, b > 0, ab > 1, vay (3) dung suy ra (2) dung.

(l + b)(i +

^

-

^

=

(21^+3)^(1 + 1)^

^

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

I

(2)

^

+ - 1 ^ vdi 1 < t < 2.
1+t
2
(31 -61^) + ( 3 1 ^ - 4 1 ^ - 9

1

1
i

•

min f (t) = f (2) = | 1 . TO do suy ra P > ^ ,
33

l
7^) >

(3)

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^

—zyxwvutsrqpo
•—

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Download Ebook Tai: https://downloadsachmienphi.com
- Phiftlng phap mien gia tri ham so.zyxwvutsrqponmlkjihgfedcbaZYXWVUT
'i--^ j^ifif '^f^'M^^'• ''• >::^r^'

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

-

Phi/dng phap lU'dng giac hoa.

-

Phi/ctng phap hinh hoc hoa.

..c.^i

,;:^^^iry,

h^> rti

'.-rl-

-

.

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

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

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

P = > y 5 - a - 2 b + V 5 - c - 2 d + N / 5 - a c - bd .

N H O N H A T C U A H A M S O T R O N G C A C KJ THI T U Y E N

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 v 5 .

'

f . i"i: V

^^
f?

Viet lai bieu iMc P dxidi dang sau:
/(a-l)2+(b-2)^

x^vlfi^m-

SINH V A O D A I H Q C , C A O D A N G
Cac bai toan tim gia trj Idn nhat va nho nha't cua ham so thu'dng xuyen xua't
hien trong cac ki thi tuyen sinh vao Dai hoc, Cao dang nhiTng nam gan day.
Trong muc nay chung toi xin gidi thieu cac bai toan ay kem theo nhffng binh
luan can thiet.

, ,/ • ^

v

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

ka-cf+ih-df

Cho

A-2011)

X,

y, z la c a c 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 "• :: -

P=
(MQ + NQ + MN) =

2x

d day CMNQ la chu vi cua tam giac MNQ.

+

3y

y+

z

z

+

x

• l;

' 1^)'?'

HUdng ddn gidi

Ta sur dung ke't qua quen bict trong hinh

Xem Idi giai trong bai toan 4, muc §1, chu'dng 1 cuon sach nay.

hoc phiing sau day: Trong cac tam giac

Binh luan:

npi tiep trong mot di/dng tron ban kinh
1. M a u chot d e giai bai 1 la d cho bang each su" dung mot ba't dang thiJc

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

CO

chu vi Idn nhat. Mat khac tam

phu, de diTa ve danh gia P > — ^ — + —

giac deu noi tiep trong du'dng tron c6

- vdi t
De'n d a y b^ng each difa vao an phu t = ^ •
y

ban kinh R c6 canh b^ng R ^/3 .
VivayCMNQ<

3N/l5.Tir(l)suyra
3^30

y .

P < ^ ^

v2

=

-N/30.

gii P

2

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

PhiTcfng phap chieu bie'n Ihien ham so.

f(t).

G 11; 2] ta quy v^ danh
'^H*>v^^-;

•

1+t

R6 rang tiep theo ta nghT ngay d e n se sur dung phiTdng phap chieu bien

. Qua 5 bai toan tren, chung toi da gidi thieu vdi c&c ban cac phiTdng phap?
PhiTdng phap bat d^ng thtfc.

+_L.
2t2+3

Do d6 maxP =

-

>

(1)

thien ham so de tim min f(t) vdi 1 < t < 2.

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 d e 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.
'

P = P(z)= —

'>;<

Cty TNHH MTV DWH Khang Vigt

Download Ebook Tai: https://downloadsachmienphi.com
Bai 2: (De thi tuyen sink Dai hoc Cao ddrtfi khoi

+ - ^ + - ^ vdiz G [l;x].

2(a^ + b^) + ab = (a + b)(ab + 2)
b^^

Tim gia tri nho nha't cua bieu thtfc P = 4

+9

2x + 3 y y + z z + x
K h i d o P'(z) = ()

L
+
X _ x ( y{y+ +z )zfiz
^ - y ( x + z)^ _
(y + z)^zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
{z + xf
+ xf
(1),

{y + zf(z + xf

fa

h\

Vb

&)

—+ — -3

a
* N e u x ^ y , lhiP(z)= ^

+^ ^ +- ^ =
= ^- V z e [ l ; x ]

5y
* Ne'u

y+ z

z+y

X > y (chii y la x > y, nen khi x

5

CO bang bien
z

1
i

P(z)

0

^

-

^

..

i

Viet lai gia thie't diTdi dang sau: 2

X
y
2x + 3y " y + ^

p ( z ) > ^ - ^ ^
2x + 3y

"

Jy

1
P = P(z)> — - — +

+
2
—

b^a b
h—

b a\
a b
+9 b^aj
ra
b

+9

b a

a

b

b

a

\

—+ —

a b^
- 1 2 —b + —
a

-2

18.

(1) •

b'
1+a J +1 = (a + b)
ab

2
'2,\/^
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
a i a j ' , ; ; : . ; J. J: - : ^ ' ^
Theo baft dang thtfc Cosi, ta c6 I + — > - p = .
ab v a b
? , ! '
!7
2
^a b "
+ — + l > ^ ( a + b) = 2V2
Thay (3) vao (2) va c6: 2
Vb ^ Va
v u ay
+
Vab
b a

^

-

—

^

.(4)

1
r —

Vay vdi mpi z e [1; x], ta c6: P(z) >

p(7^)=

a

.Hi!K

thien sau (suy ra tif (1))
1

P'(z)

b

=4 —+ —
.b
aj

y thi x > y) thi x - y > 0 nen

P'(z) = 0 < = > z ^ - x y = 0<=>z= ^ x y •
Ta

- .

b^ + a^ )

Hiidng dan giai

Difa P ve dang sau: P = 4

(x-y)(z^ - x y )

ii^.'"'

8-2011)

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

Khi do tir(4) ta c6 2(t^ - 2) + 1 > 2 > ^ t hay 2t' - 2 V2 t - 3 > 0
=>(>/2t+l)(N/2t-3)>0.

7^ + x

Do t > 2 =>

(5)

t + 1 > 0, nen tiif (5) suy ra

'

^
V2t-3>0=>t>^=>i + ^=t^-2>l.
slxzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
+yfy
^/2
b a
2

(2)

(6)

Bai toan quy ve:
•>^'-*v-'.^*'.'^^
5
Tim gia tri nho nha't cua ham so f(t) = 4t^ + 9t^ - 12t - 18 vdi t > - .

Da'u bang trong (2) xay ra o z = ^ x y .

Ta c6: f'(t) = 12t^ + 18t - 1 2 = 6(2t^ + 3t - 2) va c6 bang xet dau sau:
6 34

-2

t

Den day ta lie'p tuc giai nhiT phan sau cua bai 4, muc §1 v6i lifu y rkng do - > —
5

33

34
nen minP = —
33
Ro rang viec phat hien ra (2) theo cdch giai nay "tif n h iTron
e n " hdn
trong
Bo SGK:
each giai cua bai 4, mat du no phuTc tap hcfn ve mat tinh toan!

f'(t)

+

0

1
2
0

2

+

f(t)

Vay minf(t) = f 2 .

'4

https://bookgiaokhoa.com

23
23
= -—.ttfcmtacd P > - —
4
4

1
i

+

(7)

15

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

Download Ebook Tai: https://downloadsachmienphi.com

Da'u bkng irong (7) x a y ra khi va chi khi

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

b
_a _

b

Cty TNHH MTV DVVH Khang Vi^t

• •

,

J_

Tcrdc)

V t G [0; - 1 .

r(t)>r

,b ~ 2

a

2(a^ +b-^) + ab = (a + b)(ab+ !)<=> 2(a' + b - ) + ab = (a + b ) ( a b + l )
a > 0; b > 0
a = 2;b = l

Do f

^ 11 _ 2V3 >():=> r'(t) > 0 V t e [ 0 ; - |.
3

a > 0; b > 0
"

'

nen i d ) la hiim dong bien trcn t e |{); - ] .

" '

a = l;b = 2

. !.

^ ,

i ,, ,

Tir do suy ra 1(1) > 1(0) = 2 V t e (0; ^ ].

23
V a y minP = — ^ khi va ch'i khi a = 2, b = 1 hoac a = 1, b = 2.
Nhif the ta eo M > 2 V I e |0; ^ \
Binh

ludn:
M =2khi va ehi khi ab = be = ca; ab + be + ea = 0; a + b + c = 1 tuTe la khi va

V i c e di/a P ve dang ( I ) la Ic liT n h i c n . Cai kho la t i m m i e n xac dinh cua bien

ehi khi (a; b; e) la mot trong eac bo so ( 1 ; 0; 0), (0; 1; 0) va (0; 0; 1). D o do
gia tri nho nha't ei'ia M la 2.

b

a

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 n h i c n . D i c u do dan den vice x c t eac he thuTc (1) va (2). D e n
Bang each kc'l hdp khco leo giffa dieii kien va bat dang thiife CosizyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
la suy ra
d i c u k i c n (6). Con l a i d l nhicn la silr dung phu'dng phap ehicu bien t h i c n ham

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

so" ham so dc giai bai loan.
va suf dung phiTdng phap chieu bien thicn ham so ...