Chuy6n dg BDHSG To^n gia tr| lan nha't

g\& tr| nhd nhSt - Phan Huy Kh5i

,atBB| = x ; B , C 3 =

S + S,

De lhay A 3 B B , M , CCjMA,,

1+

•iB = A 3 B 1 ; A M

S2 = ZSj + — ,-

(3)

cung chinh la

AA3B1C2.
MB1C3

la ^

cac tam giac deu vdi canh lan liTdt la x, y, z,
cua chung thi:

(5)
>—



(1)

s/

+S^\MB,C2 +SAMA,B,

ro

up

SAA,B|C| ^ S A ^ A J C J

om

.c
ok
bo
ce

fa

w.

ww
la dien tich tam giac ay.

Tim gia trj Idn nha't ciaa

s'l +S2 + S3

CO dau b^ng trong (6) o O la trong tarn AABC.

B2C2, A3C3

^ fy^

+ y + z)

(7)

is

O la trong tarn AABC.

S

/g

O la trong tarn AABC.

--S.
3

Qua M ve ba doan thang A|B,,
CA (xcm hinh ve)

X


7S
Vay min P = — o

„

nen neu goi S,, Sj, S3 tiTdng ufng la dien tich

P = S , + S 2 + S 3 = 2 ( S , + S 2 + S 3 ) + l(sf+S^+S^)

P= — o

= C2B,

iL
ie
uO
nT
hi
Da
iH
oc
01
/

(4)

Tir (2) (3) (4) suy ra:

O


c2
\,
Sp
•S + S, =s 1 + — - + -^ = S + 2 S , + ^ = > S , - 2 S , + ^
S
s s^

Vi: ^ + ^

=z

y;C3C

(1)

DoS =

a'V3
4

minS = ^ ^
12

o

M la tam cua AABC.

tifdng iJng song song vdi AB, BC.
357

Dau bilng irong (2) xay ra

bcsin^ ^ , neu A la goc iD (90" < a < 180")
^bc, ne'u A la goc vuong ( a = 90")
NhiT vay ne'u A la goc nhon, ihi maxP = bccos^ y khi A la phan giac ngoai
cua goc A va ne'u A la goc tu, thi maxP = bcsin^ y, khi A la phan giac trong
cua g6c A va ne'u A 1^ goc vuong thi maxP = - b e khi A la phan giac trong

I

-•••^'A-

(2)
'

/g

ro

up

cua gc)c
A la dai li^ctng P dcU gia tri \&n nhat va gia tri do bSng: P| = be sin^ y.

ww

w.

fa

P = BH.CK = BH.CK'
Ap dung li luan phan 1 vao lam giac
ABC", la lhay tich BH.CK'(cung la tich
BH.CK) li'm nha'l khi A la phan giac ,
trong ciia goc BAC' (luTc A la phiin giac
ngoai ciia goc A) va gia trj Idn nhat do
= bccos — .
2

2

Theo nguycn li phan ra, thi:
maxP = max{Pi; P:} = max|bcsin^y;
358

ubccos —

B|I = C T => T B + T A + T C = B T + T I + I B , = BB|

(1)
T A + TB + TC

, „:

!NMM M IV r i W I I Kh.il"! Viet_

la d i e m T o r i c e l l i phai t i m .

(2)

C^ch3:

That vay diCng lam giac A M E sao cho E cf nufa m a t phang hci A M khong chuTa B.

^

^

:r

m-

y

Ro rang A A M C = A A R B , => M C = E B , => M A + M B + M C > T A + T B + TC


• Q u a A, B, C dirng ba

M ^ M '

Hdi/dng

T h e n tinh cha'l cua phep quay suy ra. M B M ' la tam giac d c u => M B = M M ' .
Ngoairado:

*

^tai

MC = M'A,.
(1)

up

jjnj^

s/

I

V a y M A + M B + MC = M A + M M ' + M'A,.
'' 7jy (*)

vuong

TB,


bo

Da'u bang xay ra o

cua

phep quay ta c6

goc

tao b d i M,)C va
bkng 6 0 "
=j^MoMoC = 6 0 "
B M o C = 120".

M„A|

fa

chat

w.

tinh

A M „ B = I20".

ww

.'1'VV,,,.,



T h e o bo de neu tren (ap dung vao tam giac deu QNP), ta c6:
h , + h2 + h j = T A + T B + T C
Tir do suy ra: M A + M B + M C > T A + T B + T C
(*)

•

Da'u b^ng xay ra trong (*)<=> M s T . Do chinh la dpcm.

m:(ich4:
Sis dung b6 &G h i e n nhien sau day (ChuTng m i n h ra't ddn gian va x i n danh cho
cac ban doc)
B6 de: Gia si^ A M B = B M C = C M A = 120" va M A = M B = M C t h i :
M A + M B + M C = 0

- :

Do gia thie't m a x ( A . B , C) < 120", nen t6n tai duy nhaft d i l m T trong tam giac
sao cho A T B = B T C = C T A = 120" .
160

361


Cty TIMHH MTV DVVH Khang Vi^t

ChuySn de BDHSG Toan g i i t r i Ion nhat va g i i Iri nh6 nha't - Phan Huy KhAi

'

TC

(MT +TA)TA

(Mf +TB)TB

TA

TB

= MT

TA

TB TC

TA

TB TC

r^ - o a b c a a 2a
Do vay S = — + — + - = — + — = — .

(2)

X

TC

MeBC

fa

w.

BC, CA, AB ti/dng uTng. Gia suT M la mot diem di dpng tren diTctng tron ngoai
tie'p tam giac ABC. Goi x, y, z Ian liTdt la khoang each tif M den cac canh

max M l M,)Io

MeBC

Ta c6: M,,!,, = BI„tan MoBI,, = - t a n ^ Tiif do suy ra: min S = —
MeBC

,

2a

A

=:4C0t—.

a

tan A
2
2

Cung gio'ng nhu" trong cac bai loan dai so', giai tich, v6'\c bai toan tim giii

b > c =^ 180" > A > B > C > 0

HUdngddngidi

2a

2a

MeBC

om

Nfuln xet: Da'u bang xay ra <=> M s T. Do chinh la dpcm.

p a cung CO ACLM - AABM => - = —
z
X

s/

MB.TB

up

TA

•+

(1)

Tir(l)(2)=^^ +^ =- ^ l i l ^ =i .
y z
X
X

2. a.p
(1)

. Vav lir (6) (7) di den: min S = 4cot— 44. M la trung diem cua BC


Ctiuygn de BDHSG Tcrin gia

tf| I6n nha't

g\i

trj nh6 nliil

Cty TNHH MTV DVVH Khang Vi?t

Phan Huy KhJi

Bai 6. Cho hlnh tron ban kinh r. Xet ta't ca cac tvt giac ABCD ngoai tiep dtfcj^
tron. Tim gia trj nho nhat cua dai luTOng P = AB + CD.
,
Hitdng ddn gidi
Goi M la tam dtfSng tron npi tiep ti? giac => M
giao diem cua cac diTdng phan gi^c trong cua cac
goc A, B, C, D cua ti? giac.
Ve di/dng tron ngoai tiep tam giac ABM va gpi
ABN la tam giac can npi tiep c6 dinh la N sao
cho ANB = AMB.
Gpi h la khoang each tif N xuong AB, con h, la
khoang each tir M xuong AB. Khi do ta c6:
h > hi va hi = r.

Dau bkng trong (3) xay ra o dong thcfi c6 dau bkng trong (1) (2).
Taco: AMB-f CMD = 1 8 0 " - A l l + 180" - C + D = 180^'
tan AMB = cot CMD
tan

1 - . Vi theo bat dang thiJc Cosi, suy ra:
CMD

1 H-tan
^ AMB
CMD
tan
1-tan
tan CMD
364

iL
ie
uO
nT
hi
Da
iH
oc
01
/

Ti/Png ixi ta c6: CD > 2r t a n ^ ^ .

Ta
1. Khi do ta c6: AB < 1, BC < 1, CD < 1.
Dat AC = X va gpi M la trung diem cua AC
Ta eo: A B ' + B C ' = 2BM^ + AC^

>2.

(4)

a + 3 = 45"

tan((v + 3) = 1

:A *

tan (x + t a n 3

,
x+ y
=
1=^
1 - x y^ = 1
l - t a n a + tan3
>x
Ta

+ y = 1 - xy.
c6:

_ ^

X

y

~

'

gia tri Idn nhat va nho nha't ciia S.

A B C D la nii-a luc giac deu canh 1.

Taco: AC + A D = a

72) = d ( l + 72).

A B va C D sao cho M B N -=45*'. Gia sijf S la di?n tich tarn giac B M N . T i m

(10)

HUdng ddn gidi

+

i 9. Cho hinh vuong A B C D canh bang 1. D i e m M va N Ian liTdt di dpng tren

.c

o

+ AB = A B ( I

72

Gici trj nho nha't dat diTdc khi A B la diTdng cheo cua hinh vuong.


-o

72

AB

Ta

AB = BC =

Ket hcfp l a i ta c6: S =

/ = AC + A D + A E +

.

x ' = 6 - x^ <=> x = N/3 .
/3

I

AB

do

TiJf (1) ta c6: A F > A B , nen ket hdp l a i suy ra

(9)

Tir do ket hcJp v d i (5) co: S < — . 7 3 = 4

hi
Da
iH
oc
01
/

T v / - / ; \ / ' 7 \x

AB
TiS do suy ra: A C + A D = a > -

(6)

2

ta c6:

[x + y

SACM - SBCN -

(l-x)(l-y) ^

S

=

1-t


Tir do suy ra:
AB
'

Ap dung dinh l i Viet vdi (3) ta c6: x, + X2 = I - t; x,X2 = t nen
t-(l-t) + l>0

242 2

2 + V2

hay V 2 - 1 < S < ^ .

',

X|X2>0

>0

X, +X2

Tir (5) (6) suy ra:

X | X 2 - ( X | + X 2 ) + 1>0

R i ( l + cosa) = Rcosa =:> R| = Rcosa
(2)
1 + cosa
Hoan to^n ti/dng tiT, ta c6 R2 =

(6)

(1)

T i r ( l ) ( 2 ) (3) suy r a / =

Rsina

(3)

1 + sina
Rcosa sina

Rsina cosa

1 + cosa cosa

1 + sina sina
369


l + cosa

•H

+COS

2R

-.
s i n a + cosa + 1

L a i Iheo ba't dang thuTc Cosi, thi

R

A
B
B
C , C,
A ^ . J
2A , 2B. 2C
1 = t a n — t a n — + tan—tan — + tan —tan — > 3 : V t a n — t a n —tan —

a ( . a
cos
sin + cos
2 I
2
2

a
2

2 2
0 vdi m o i tarn giac A B C . T i f do ta c6 maxP = VmaxP^

, ^2

Taco

2A
B
C I ,
. 2A
P = cos —cos—cos—= — 1 - sin —
2

ce

tan^ — + tan^ — > 2 t a n — t a n —
2
2
2
2
. 2B
2C ,
B
C
tan^ — + t a n ' ' — > 2 t a n — t a n —
2
2
2

Docos^—^

Ta

V a y min/ = 2 R ( N/2 - 1), k h i C la di em chinh giffa cua A B .

Tir do suy ra:

B
C
cos —cos— d day A , B, C
2
2

B a i 2 . T i m gia trj Kin nha'l cua bicu thuTc P = '^"'^

<?>C\A triing d i e m ciia A B

2

2

26
Nhi/vay minP = — o A B C la tarn giac deu.

/ = 2 R ( N ^ - 1 )
o A = B = C.

N/2+1

2

V

Da'u b^ng trong ( 4 ) xay ra o dong thcJi c6 da'u bang trong (2), (3)

• = 2R(N/2-1)

tan^ ~ + tan^ — + tan^ — >

2

26
Da'u bang trong (3) x a y ra o A = B = C. T i f (2) (3) c6 P > — .

=> 0 < sina + cosa < yfl ,\\e tiT (3) c6 :

*

2

2A
2B
2C
1
tan'' — tan —tan — < —


o B =C

A\ ,
• A
2-2sin1 + sin —
2 A
2

. • A'
2-2sn I
2>

• A—
1 + sin
I

(.

• A ^ (,
• A^
+ I 1 + sin 2 ; + I 1 + sin 2)

n3

4

,116

'27

<=> A B C la tarn giac can dinh A v d i A = 2arcsin ^ .

\,

'ii

B a i 3. X e t cac tarn giac A B C thoa man he thtfc tanA + tanC = 2tanB.

gidi

I

cosB = 2cosAcosC (do sinB = sin(A + C)^Q)

=>
=>

cosB = cos(A + C) + cos(A - C)
2C0SB = C O S ( A - C).

,

(1)

s/

Tur (3) suy ra S > 0, nen m a x S = V m a x S ^ .

.



= B = C.

/g

o A
9

T i r ( l ) ( 4 ) CO

S =i o A =B =C
9
1
V a y minS = ^

fa

;

D a u bang trong (4) xay ra o dong thcfi c6 dau b i n g trong (2) (3)

la tarn giac deu.

w.

B a i 5. X c t cac tarn giac A B C v d i A la goc Idn nha't.

ww

T i m gia t r i Idn nhat cua dai lifdng S = sin2B + sin2C + sin A

2

x-" + y + z >

(4)

V i the theo bat dang thiJc Co si di den

CO

om

.c

(3)

2

Vx, ^ y .

+ y + z)(x' + y ' + z') > (x^ + y^ + v?)^

ro

(2)

Tir(l)(2)suyraS=2sin|^i±^.

TiT do ta


A
B
B
C
C
A
D a t X = t a n y t a n - j , y = tan — t a u y , z = t a n y t a n y thi x + y + z = 1

..^^

=>

• ( 2 - 2 c o s B ) + (l + 2cosB)

ddn gidi

Apdungba'tdangthuTcBunhiacopskichohaiday

sin(A + C)
2sinB
—
=
cos A cos C
cosB

Taco S ^ = 4 s i n 2 - i i ^ ^ =
2
2

'


3
B = arccos—
4

• ,

, j

f

Do sinA > 0 va cos(B - C) < 1, nen lif (1) cd:
S < 2sinA +

2
sin A

(2)
373


Chuyfin 66 BDHSG Toan gii trj Idn nha't vi giA trj nh6 nhat - Phan Huy Kh&\

Cty TIMHH MTV DWH Khang ViSt

Da'u bang trong (2) xay ra <=> cos(B - C) = 1 <=> B = C.

Dau bang trong (4) xay ra

Do A la goc l(^n nhii't trong lam giac => A > ^ => ~
45"

X c l ham so i\x) = 2\ - wYi —
1

S
(1) di/cJc chiJng minh! '

Tir do ta CO
Thay (3) vao (2) ta di den P < c o l ' - + c o l 2
2
'

cos(B + C) > 0.

ok

Bai 6. X e l tarn giac ABC vdi A > 90".

Tr\idc hct ta c6 tanBtanC < cot" A

' u ;

om

<=> ABC la tarn giac dcu.

P - l o B - C v a c o t —= 1
2
« ABC la tarn giac vuong can tai A

.c

7V3

f(x)

-

s/

f'(x)

0 < cot — < 1). *


Chuy6n dg BDHSG Toan gia Irj I6n nhS't vi gia Irj nhd nha't - Phan Huy KhJi

Cly TNHH MTV DWH KhangVi§t

Bai 7. Xet tap hcJp cac tarn giac ABC khong phai la tam giac c6 goc tu.

Tir do C O bang bien thien sau:

Tim gia tri nho nha't cua da i liTrtng P ^ sin A + sinB + sinC
cos A + cosB + cosC
Hudng ddn gidi
Do vai tro binh dang cua A, B, C nen c6 the gia suf A = max{A; B; C}
71

.

3

X

^=
2
2
2

(0 — + 1 .

3A

7t ^

min g(A) = g

2"

C

Datx=cos

I +

2
cot — + cot — + cot —.
Z

Z

Zr

(1)
.

Dau bang trong (1) xay ra <=> A = B = C.
A
B

2
2
2
377


ChuySn dg BDHSG 7oAn gia Irj Idn nha't

gia trj nh6 nhU't - Phan Huy Khii

!S/hdn xet: Xet cac each giai khac sau day:
1. Xet cac vectd ddn vi e , , C 2 , C 3 nhtfhinhve.
Ta
Xethams6'f(x) = x + 1

CO

(e, + 62

f'(x) = l - ± , nen c6 bang bien thien sauc,

3v^

f'(x)
f(x)

+00

1
1

ro

d day A, B, C la ba goc ciia tarn giac ABC.

/g

Hitdng ddn gidi

om

S = 2cos^^-tlcos^—^ + l - 2 s i n 2 2
2
2
^ . C
A-B , ^ . 2C
= 2sm—cos
t-l-2sin —

.c
ok

w.

•2A-B

-sir-—-+3

Tir(l)suyra

3


Tifdng ty A = C = 6O" => ABC la tam giac deu.

Neu x, y e

fa

4sin^cos^^—^ + 2 - 4 s i n 2 2„, ,,,,,2„.,::„,.,,.,„f
2

MBP = 120" o

Ta CO nhan x6t hien nhien sau:

bo

2

o

Xet each giai khdc nffa nhU'sau:

ce

2

^v,

o BMNP la hinh thoi c6 B M = BN = BP = 1



2
• A-B „
sm
= 0
T .

C

A-B

2 .

. 2

2sin — = cos
Vay maxS = -

,|cos(e,,e3) + 2|t

Do

X = 3yf3

2 ,

cos(e|,e2)

Do cos (c|, 62) = cos (18()" - B j = - cos B . -


01
/

X

+63)^

=>
o A = B = C.

cosA + cosB + cosC + IT 1
^
—2- < cos— = — => cosA + cosB + cosC < —.
4
~
3 2
2

Dafu bkng xay r a < » A = B = C = ^ .

ABC la tarn giac deu.
379


--

V a y maxS = -

«


xay ra o

.

A-B

= 1 o A

3

§1. V a i bai loan m 6 dau

+ cosC < 2sin — + cosC
2

cos

1. M('/ d i u v6 gia tri l«tn nha't va nho nhS't cua ham s6'

3

§2. N h i n l a i cac bai loan ve gia t r i Idn nha't va nho nha't cua h a m so trong
cac k l t h i t u y e n sinh vao dai hoc, cao dS^ng

= B

13

• '^i;'



(do 0 < X < j i )

V d i chu y k h i 0 < X < j i t h i cos J > 0, ta c6 bang b i e n thien sau:

3

om

suy ra S < - , va S =
2

2

la tam giac deu.


ChiTcTng 4. Phi/dng phap c h i ^ u bien thidn h a m s6' tim gia trj Idn nhd't

w.

~ 3

1.2. Phirdng phap suT diing trifc tiep bat dang thiJc Cosi

Fhifrfng phap Ivhfng giac h6a tim gia trj Idn nhS't va nho nhfl't c u a h a m s6'

fa

ABC

42

Chrfdng 3.

ce

A = B

42

1.1. Sijrdung ba't ding thuTc Cosi c d ban

v^ nho nha't cua h a m so

ok


ro

1

f(x)

+

•'

/g

1

f'(x)

n

up

0

Ta

IT
X

30

hiidng 2. Phrfefng phap suT dyng bS't dang thrfc de tim gia trj Idn nhSft va nho nhS't

va n h o nhS't m a c h u n g t d i d a t r i n h
that phong phii l a m sao!

Chifdng 6. Phrfdng phap siJ dung 66 thj hoSc hinh hqc
d4 tim gia trj idn nhS't va nho nhS't cua h a m so'

243


C h i f r f n g 7 . L f n g d u n g c u a g i a t r f l«?n nha't v a g i a t r j n h o n h a t t r o n g b a i t o a n
g i a i phi/(fng t r i n h v a bS't phi/«/ng t r i n h c o t h a m .s(Y

263

§ 1 . M o ' i l i e n h e giiJa g i a t r i i d n n l i a l , n h o nha't c u a h a m so
•

v a sU C O n g h i p m ci5a p h i f i t n g t r i n h v a b a t phU(

'

308

Ta

p h u t h u o c t h a m so'