K12_GT_CI_LT Ung Dung Dao Ham

TI LIU TNG HP

NG DNG O HM

Kho st v v th hm s LUYN THI THPT QUC GIA

BI DNG HC SINH GII

HU, 2015

1

LI NI U

Vi mong mun gip cc em hc sinh thc hin tt qu trnh t hc,

t nghin cu v chun b cho k thi Trung hc ph thng Quc gia, ng

thi cng h tr bn thn trong qu trnh ging dy mn Ton 12 v

bi dng hc sinh gii, chng ti tin hnh tng hp tp ti liu ng

dng o hm kho st v v th hm s Luyn thi THPT Quc

gia v bi dng hc sinh gii.

Tp ti liu c chia lm hai Phn chnh, hai Ph lc v Hng dn

gii p s, trong Phn 1 gm su chuyn c bn:

Chuyn 1. S bin thin ca hm s.

Chuyn 2. Cc tr ca hm s.

Chuyn 3. Gi tr ln nht ca hm s.

Chuyn 4. ng tim cn ca th hm s.

Chuyn 5. Kho st s bin thin v v th hm s.

Chuyn 6. Mt s bi ton lin quan kho st hm s.

Phn 2 gm ba chuyn nng cao:

Chuyn 1. S bin thin Cc tr v Gi tr ln nht, gi tr nh

nht ca hm s;

Chuyn 2. Tim cn v s tng giao ca hai th hm s.

Chuyn 3. Mt s bi ton khc.

Hai ph lc vi ni dung tng ng:

Ph lc 1. Kho st hm s v cc bi ton lin quan qua cc k thi

i hc Cao ng (t 2002 n nay).

Ph lc 2. S dng MTCT gii mt s bi ton phng trnh v h

phng trnh.

V cui cng l Hng dn gii v p s.

2

Mi chuyn bao gm nhiu vn khc nhau, u c bt u

bng cch nu Phng php, tip theo l mt s V d mu v cui cng

l Bi tp tng t.

Kin thc trong tp ti liu hon ton khng c g mi m; chng ti

ch tng hp v sp xp li theo ca mnh. Tp ti liu c hon

thnh nh vo qu trnh su tm, ch bn t ngun ti liu phong ph, a

dng nh sch, bo, tp ch, t bit l t internet, kt hp vi kinh

nghim ca chng ti trong qu trnh ging dy cc lp n thi i hc

Cao ng v bi dng hc sinh gii. Mi sai st ngi tng hp xin nhn

trch nhim v li khng nhn thy c trong qu trnh ch bn.

Chng ti xin by t lng bit n chn thnh n qu bn b ng

nghip c v cho kin v tp ti liu ny cng nh cm n s chia s

ti liu qu bu ca qu thy, c thng qua mng internet.

Mc d c nhiu c gng, nhng nhng thiu st l kh trnh khi,

chng ti rt mong nhn c s gp ca qu v tp ti liu c

hon chnh hn.

Vi xu hng ci cch gio dc hin nay, tp ti liu ny cng ch s

dng c thm vi nm na!

Ma h nm t Mi 2015

Ngi tng hp: Trn Quang Thnh

3

MC LC LI NI U .................................................................................................................................. 1

PHN 1. MT S VN C BN .......................................................................................... 7

CHUYN I. S BIN THIN CA HM S ....................................................................... 9

VN I. S BIN THIN CA HM S KHNG C THAM S ............................. 11

PHNG PHP. ............................................................................. 11

V D MU. .................................................................................... 11

BI TP TNG T. ........................................................................ 14

VN II. S BIN THIN CA HM S C THAM S ....................................... 15

PHNG PHP. ............................................................................. 15

V D MU. .................................................................................... 15

BI TP TNG T. ........................................................................ 17

VN III. PHNG PHP HM S TRONG CHNG MINH BT NG THC ........ 19

PHNG PHP. ............................................................................. 19

V D MU. .................................................................................... 19

BI TP TNG T. ........................................................................ 21

VN IV. PHNG PHP HM S GII PHNG TRNH, BT PHNG TRNH V

H PHNG TRNH .......................................................................................... 23

PHNG PHP. ............................................................................. 23

V D MU. .................................................................................... 23

BI TP TNG T. ........................................................................ 28

CHUYN II. CC TR CA HM S ................................................................................ 29

VN I. CC TR CA HM S KHNG C THAM S ....................................... 31

PHNG PHP. ............................................................................. 31

V D MU. .................................................................................... 31

BI TP TNG T. ........................................................................ 34

VN II. CC TR CA HM S CHA THAM S ............................................. 35

PHNG PHP. ............................................................................. 35

V D MU. .................................................................................... 37

BI TP TNG T. ........................................................................ 41

CHUYN III. GI TR LN NHT V GI TR NH NHT CA HM S .............. 43

4

VN : NG DNG O HM TM GTLN, GTNN ............................................. 43

PHNG PHP. .............................................................................. 43

V D MU. .................................................................................... 44

BI TP TNG T ......................................................................... 47

CHUYN IV. NG TIM CN ....................................................................................... 49

PHNG PHP. .............................................................................. 49

V D MU. .................................................................................... 50

BI TP TNG T. ........................................................................ 52

CHUYN V. KHO ST S BIN THIN V V TH HM S ........................... 53

PHNG PHP. .............................................................................. 53

V D MU. .................................................................................... 54

BI TP TNG T. ........................................................................ 56

CHUYN VI. MT S BI TON LIN QUAN KHO ST HM S ......................... 57

VN I. S TNG GIAO CA CC TH ................................................... 57

PHNG PHP. .............................................................................. 57

V D MU. .................................................................................... 57

BI TP TNG T. ........................................................................ 59

VN II. BIN LUN NGHIM PHNG TRNH ............................................. 61

PHNG PHP. .............................................................................. 61

V D MU. .................................................................................... 62

BI TP TNG T. ........................................................................ 66

VN III. BI TON TIP XC GIA CC NG .......................................... 67

PHNG PHP. .............................................................................. 67

V D MU. .................................................................................... 69

BI TP TNG T. ........................................................................ 73

VN IV. MT S BI TON LIN QUAN IM C BIT TRN TH HM S . 75

PHNG PHP. .............................................................................. 75

V D MU. .................................................................................... 77

BI TP TNG T. ........................................................................ 81

PHN 2. MT S VN NNG CAO.................................................................................. 83

CHUYN I. ............................................................................................................................. 85

5

S BIN THIN - CC TR V GI TR NH NHT GI TR LN NHT CA HM

S.................................................................................................................................................. 85

V D. ........................................................................................... 85

BI TP. ........................................................................................ 90

CHUYN II. ........................................................................................................................... 93

TIM CN V S TNG GIAO CA HAI TH HM S ........................................... 93

V D. ........................................................................................... 93

BI TP. ..................................................................................... 102

CHUYN III. MT S BI TON KHC ...................................................................... 105

V D. ........................................................................................ 105

BI TP. ..................................................................................... 106

PH LC 1. KHO ST HM S V MT S BI TON LIN QUAN QUA CC K

TUYN SINH I HC CAO NG TON QUC ........................................................ 109

PH LC 2. S DNG MY TNH CM TAY GII MT S BI TON PHNG

TRNH V H PHNG TRNH ......................................................................................... 131

I. PHNG TRNH QUY V PHNG TRNH BC 4 ......................................... 131

V D MU. ................................................................................. 131

BI TP TNG T. ..................................................................... 135

II. H PHNG TRNH A THC ................................................................. 135

V D MU. ................................................................................. 135

BI TP TNG T ...................................................................... 137

III. ON NGHIM V DNG LNG LIN HP GII PHNG TRNH ................ 137

V D MU. ................................................................................. 137

HNG DN GII P S ............................................................................................... 141

PHN I. MT S VN C BN ............................................................... 141

PHN 2. MT S VN NNG CAO .......................................................... 153

TI LIU THAM KHO ......................................................................................................... 163

7

PHN 1. MT S VN

C BN

9

CHUYN I. S BIN THIN CA HM S

1. nh ngha.

Hm s ng bin (tng) trn

( ) ( )

Hm s f nghch bin (gim) trn

( ) ( )

2. iu kin cn.

Gi s c o hm trn khong .

a) Nu ng bin trn khong th ( )

b) Nu nghch bin trn khong th ( )

3. iu kin .

Gi s c o hm trn khong .

a) Nu ( ) ( ( ) ti mt s hu hn im) th

ng bin trn

b) Nu ( ) ( ( ) ti mt s hu hn im) th

nghch bin trn .

c) Nu ( ) th khng i trn .

Ch : Nu khong c thay bi on hoc na khong th phi lin

tc trn .

11

VN I. S BIN THIN CA HM S KHNG C

THAM S

PHNG PHP.

xt chiu bin thin ca hm s ( ), ta thng thc hin cc

bc nh sau:

+ Tm tp xc nh ca hm s.

+ Tnh y . Tm cc im m ti y = 0 hoc y khng tn ti (gi l

cc im ti hn)

+ Lp bng xt du y (bng bin thin). T kt lun cc khong

ng bin, nghch bin ca hm s.

V D MU.

Bi mu 1.1.1. Tm cc khong tng gim ca cc hm s sau

a.

. b. ( )

c. d.

2 2.

2

x xy

x

e.

5 1.

2 3

xy

x f. .

g. h.

2

1.

1

xy

x x

GII. a. Tx . Ta c hoc .

Bng bin thin:

0 0

12

Vy, hm s ng bin trn cc khong ( ) ( ) hm s

nghch bin trn ( )

b. Tx Ta c hoc

Bng bin thin:

0 0 0

Vy, hm s ng bin trn cc khong ( ) ( ) v nghch

bin trn cc khong ( ) ( )

c. Tx . Ta c hoc .

V ( ) ( ) nn ta c bng bin thin:

0

Vy, hm s ng bin trn ( ) v nghch bin trn ( ).

d. Tx Ta c

( )

Bng bin thin:

2 4 0

13

Vy, hm s ng bin trn cc khong ( ) ( ) v nghch bin

trn cc khong ( ) ( ).

e. Tx

2

3\ . Ta c

2

2' 0, .

32 3

17y x

x Do , hm s ng

bin trn cc khong

2 2; ; ; .

3 3

f. Tx [ ] Ta c

Bng bin thin:

0 1 2 0

Vy, hm s ng bin trn ( ) v nghch bin trn ( )

g. Tx [ ] Ta c

Bng bin thin:

1 3 5 0

Vy, hm s ng bin trn ( ) v nghch bin trn ( ).

g. Tx Ta c

( )

( )

14

Bng bin thin:

1 0

Vy, hm s ng bin trn ( ) v nghch bin trn ( )

BI TP TNG T.

Bi tp 1.1.1. Xt chiu bin thin ca cc hm s sau:

a. 2(4 )( 1)y x x b. 4 21 1

210 10

y x x c.

2 1

5

xy

x

d.

11

1y

x e. 4 3 26 8 3 1y x x x f.

2

2

1

4

xy

x

g. 3 2 2y x x h. 2 1 3y x x i. 22 .y x x

15

VN II. S BIN THIN CA HM S C THAM S

PHNG PHP.

Cho hm s ( , )y f x m , l tham s, c tp xc nh .

Hm s ng bin trn D 0, x D.

Hm s nghch bin trn D x D.

T suy ra iu kin ca m.

NHC LI. Nu th:

0

0' 0,

0

0

a b

cy x R

a

0

0' 0,

0

0

a b

cy x R

a

V D MU.

Bi mu 1.2.1. Cho hm s ( ) . Tm

a. Hm s ng bin trn

b. Hm s ng bin trn (0; ) .

c. Hm s ng bin trn (1; ) .

d. Hm s nghch bin trn khong c di ng bng 1.

GII. Vi tp xc nh ta c

a. Hm s cho ng bin trn khi v ch khi vi mi

. iu ny tng ng vi hay .

b. Hm s cho ng bin trn ( ) khi v ch khi vi mi

( ) hay , vi mi ( )

t ( ) ( ).

Hm lin tc trn nn lin tc trn ( )

16

Ta c ( )

Bng bin thin ca :

-1 0 + 0 - -

0

Qua bng trn ta thy rng, ( ) trn ( ) khi v ch khi

.

Vy, hm s ng bin trn ( ) khi v ch khi .

c. Hm s cho ng bin trn ( ) khi v ch khi vi mi

( ) hay , vi mi ( )

t ( ) ( ).

Hm lin tc trn nn lin tc trn ( )

Ta c ( )

Bng bin thin ca :

-1 1 + 0 - -

Qua bng trn ta thy rng, ( ) trn ( ) khi v ch khi

. Vy, hm s ng bin trn ( ) khi v ch khi .

d. Xt c

Nu th vi mi nn hm s cho

ng bin trn ; trng hp ny khng tha mn yu cu bi ton.

Nu th c hai nghim phn bit Lc ,

v h s ca l nn hm s nghch bin trn ( ). Nh vy,

hm s nghch bin trn on c di bng 1, ta cn thm

17

| |

| |

( )

Vy, hm s gim trn on c di bng 1 khi v ch khi 9

.4

m

BI TP TNG T.

Bi tp 1.2.1. Cho hm s 3 21

( 1) (3 2)3

y m x mx m x (1). Tm tt c

cc gi tr ca tham s m hm s (1) ng bin trn tp xc nh ca

n.

Bi tp 1.2.2. Cho hm s 3 23 4y x x mx . Tm tt c cc gi tr ca

tham s m hm s cho ng bin trn khong ( ;0) .

Bi tp 1.2.3. Cho hm s 3 2(1 2 ) (2 ) 2y x m x m x m . Tm m

hm ng bin trn khong (0; )K .

Bi tp 1.2.4. Cho hm s 3 23y x x mx m (1), (m l tham s). Tm

m hm s (1) nghch bin trn on c di bng 1.

Bi tp 1.2.5. Cho hm s 4 22 3 1y x mx m (1). Tm m hm s

(1) ng bin trn khong (1; 2).

Bi tp 1.2.6. Cho hm s

4mxy

x m (1). Tm tt c cc gi tr ca tham

s hm s (1) nghch bin trn khong ( ;1) .

19

VN III. PHNG PHP HM S TRONG CHNG

MINH BT NG THC

PHNG PHP.

Trong vn ny, chng ti ch xt n vi bt ng thc mt bin

n gin. Vic vn dng hm s trong chng minh cc bt ng thc

nhiu bin s c trnh by trong chuyn nng cao Gi tr ln nht,

nh nht ca hm s thuc tp sch ny.

chng minh bt ng thc (mt bin) bng phng php hm s

ta thc hin cc bc sau:

Chuyn bt ng thc v dng ( ) (hoc

20

GII.

a. Xt hm s ( ) lin tc trn na khong

0;

2 v c

o hm ( ) vi mi thuc

0; .2

Do hm s cho ng bin trn

0;

2 v ta c

( ) ( ) (

)

(

)

Hin nhin

Vy vi mi .

b. Xt hm s 2

( ) cos 12

xg x x lin tc trn [ ) Theo cu a), ta

c ( ) vi mi nn hm ng bin trn

[ ) Do , vi mi ta c ( ) ( ) hay

Khi ta c( ) v do

( ) ( )

Vy, 2

cos 12

xx , 0x .

c. Xt hm s ( ) lin tc trn

0;

2 v c o hm

( ) (

)

21

Do hm s cho ng bin trn

0;

2 v ta c

( ) ( ) (

)

(

)

d. Xt hm s tan

( )x

f xx

lin tc trn

0;2

v c o hm

( )

(

)

Ta s chng minh ( ) vi mi

0;2

x bng cch chng minh

rng ( ) vi mi

0; .2

x

Tht vy, ( ) l hm lin tc trn

0;

2 v ( )

vi mi

0;2

x nn hm ng bin trn

0;2

, do

( ) ( ) (

)

T suy ra hm s

( )

(

)

( ) ( )

BI TP TNG T.

Bi tp 1.3.1. Chng minh cc bt ng thc sau:

a)

2 1

sin tan ,0 .3 3 2

x x x x b)

sin tan 2 , 02

x x x x .

22


a)

sin sin ,0 .2

a a b b a b

b)

tan tan , 0 .2

a a b b a b


a)

2sin ,0 .

2

xx x

b) 3 3 5

sin , 0.6 6 120

x x xx x x x


a) 0tan55 1,4. b) 01 7

sin20 .3 20

c) 2 3log 3 log 4 .

23

VN IV. PHNG PHP HM S GII PHNG

TRNH, BT PHNG TRNH V H PHNG TRNH

PHNG PHP.

1. Dng 1: S dng tnh cht ca hm s lin tc.

p dng nh l: nu hm s ( ) lin tc trn [ ] v c

( ) ( ) th tn ti ( ) sao cho ( ) . Ngha l phng

trnh ( ) c t nht mt nghim trn ( ).

Phng php ny thng s dng chng minh phng trnh c t

nht mt nghim tha mn iu kin no .

2. Dng 2: S dng tnh n iu i vi hai hm s c chiu bin

thin ngc nhau.

Chuyn phng trnh v dng ( ) ( ) ( ), trong l hm

ng bin trn v l hm nghch bin trn .

T suy ra th hai hm s ct nhau ti nhiu nht mt im, hay

phng trnh ( ) cho c nhiu nht mt nghim trn .

Nhm mt nghim no ca ( ) th y l nghim duy nht.

3. Dng 3: S dng tnh n iu i vi mt hm s.

S dng nh l: nu l mt hm tng (hoc gim) trn ( ) th

( ) ( ) ( )

V D MU.

Bi mu 1.4.1. (D 2004) Chng minh rng phng trnh

c ng mt nghim thc.

GII. Phng trnh c vit li ( )

24

Nhn thy rng, nu th v ( ) nn lc ny

phng trnh v nghim.

Nu th ta xt hm s ( )

Ta c ( ) ( ) ( ) vi

mi ng thi lin tc trn [ ) nn ng bin trn

[ ) Do phng trnh ( ) c nhiu nht mt nghim trn

[ )

Mt khc ( ) ( ) nn ( ) c t nht mt nghim trn

[ )

Vy phng trnh c ng mt nghim trn [ ) nn phng trnh

c ng mt nghim trn

CCH KHC. Phng trnh cho tng ng vi phng trnh ( )

Nhn thy khng phi l nghim phng trnh nn ta vit li

nh sau, vi iu kin

(

)

(

)

Trn ( ) hm s ( ) ng bin v hm s 1

( ) 1g xx

nghch bin nn phng trnh trn c nhiu nht mt nghim trn

khong ny.

n y, l lun tng t cch trn, ta c iu cn chng minh.

Bi mu 1.4.2. (C 2012) Gii phng trnh

( )

GII.

iu kin 1

.2

x Phng trnh tng ng vi

25

( ) ( ) ( )

Xt hm s ( ) Ta c ( ) vi mi

. Suy ra hm s ( ) ng bin trn . Do

( ) ( ) ( )

Vy, tp nghim phng trnh l

{

}

Bi mu 1.4.3. Gii h phng trnh

{( ) ( ) ( )

( )

GII. iu kin v .

Ta c | | do

Mt khc ( )( ) nn t phng trnh

( ) ta c ( )

Xt hm s ( ) . Ta c

( )

nn hm ng bin trn do ( ) ( ) ( )

Cch 1.

Thay vo phng trnh ( ) ta c

[ (

)] [ (

)]

26

( ) [

] ( )

V ( ) v

( )

nn ( ) hoc .

Vy, nghim h phng trnh l v .

Cch 2. Thay vo phng trnh ( ) ta c

t ( ) [ ]

( )

( )

( )

( )

( ) ( )

V ( ) vi mi ( ) nn hm s ( ) ng bin

trn ( ). Do phng trnh ( ) c nhiu nht mt nghim

trn ( ). (5)

Mt khc, hm lin tc trn [ ] v

( )

( )

do ( ) c t nht mt nghim trn ( ). (6)

T ( ) v ( ) suy ra ( ) c ng mt nghim ( ).

Bng bin thin ca

( )

( )

27

T bng bin thin ta thy rng phng trnh ( ) c nhiu nht

hai nghim trn ( ). M v l hai nghim ca ( )

nn y l hai nghim cn tm.

Vy, nghim h phng trnh l v .

Bi mu 1.4.4. Gii h phng trnh

{ ( ) ( )

( )

GII. T phng trnh ( ) ta c

( ) ( )

{

Thay vo phng trnh ( ) ta c

( )

t ( )

( )

( )

V nn v

Do ( ) vi mi v lin tc trn [ ) nn ng

bin trn [ ). Suy ra phng trnh ( ) c nhiu nht mt nghim

dng. Nhn thy 1

2x l nghim ca ( ) nn y l nghim duy nht

ca ( ) Vy, h cho c nghim 1

; 1.2

x y

28

BI TP TNG T.

Bi tp 1.4.1. Gii cc phng trnh sau:

a. 3 1 7 2 4x x x . b. 24 1 4 1 1x x .

Bi tp 1.4.2. Gii cc phng trnh sau:

a. 3 32 23 32 1 2 1 2x x x x . b. 3

1 2(4 ) 2x x

x x.

Bi tp 1.4.3. Gii cc bt phng trnh sau:

a. 2

1 1 24

xx x . b. 5 5(1 ) (1 ) 4 2x x .

Bi tp 1.4.4. Gii cc h phng trnh sau y:

a.

2

2 2

4 3 5 2 0,

4 3 4 7.

1

2

x x y y

x xy

b.

3

3 2 2 2 1 0,

2 2 2 1 1.

x x y y

x y

Bi tp 1.4.5. Chng minh rng phng trnh 5 2 2 1 0x x x c ng

mt nghim.

Bi tp 1.4.6. Tm phng trnh 2 2 4 (3 )( 1) 3x x x x m

lun c nghim.

29

CHUYN II. CC TR CA HM S

I. Khi nim cc tr ca hm s.

Gi s hm s xc nh trn tp ( ) v

a) im cc i ca nu tn ti khong ( ) v

( ) sao cho ( ) ( ), vi mi ( ) .

Khi ( ) c gi l gi tr cc i (cc i) ca .

b) im cc tiu ca nu tn ti khong ( ) v

( ) sao cho ( ) ( ), vi mi ( ) .

Khi ( ) c gi l gi tr cc tiu (cc tiu) ca .

Cc i v cc tiu ca hm s c gi chung l cc tr ca hm

s.

c) Nu l im cc tr ca th im ( ( )) c gi l

im cc tr ca th hm .

II. iu kin cn hm s c cc tr.

Nu hm s c o hm ti v t cc tr ti im th

( )

CH .

+ o ca nh l ny khng ng, ngha l ( ) th cha chc

hm t cc tr ti .

V d: Xt hm s ( ) .

+ Hm s ch c th t cc tr ti nhng im m ti o hm

bng 0 hoc khng c o hm.

+ Nu hm c o hm ti v c cc tr ti im th tip tuyn

ca th hm s ti im 0 0 0; ( )M x f x song song vi trc honh .

30

III. iu kin hm s c cc tr.

1. nh l 1. Gi s hm s lin tc trn khong ( ) cha im

v c o hm trn ( ) .

a) Nu ( ) i du t m sang dng khi i qua th t

cc tiu ti .

b) Nu ( ) i du t dng sang m khi i qua th t

cc i ti .

2. nh l 2. Gi s hm s c o hm trn khong ( ) cha im

( ) v c o hm cp hai khc 0 ti im .

a) Nu ( ) th t cc i ti

b) Nu ( ) th t cc tiu ti .

31

VN I. CC TR CA HM S KHNG C THAM S

PHNG PHP.

Ta thng s dng hai quy tc sau y:

QUY TC 1. Dng nh l 1.

Tm ( )

Tm cc im ( ) m ti o hm bng 0 hoc

khng c o hm.

Xt du ( ) Nu ( ) i du khi i qua th hm s t

cc tr ti .

QUY TC 2. Dng nh l 2.

Tnh ( )

Gii phng trnh ( ) tm cc nghim ( )

Tnh ( ) v ( ) (i = 1, 2, ).

Nu ( ) th hm s t cc i ti .

Nu ( ) th hm s t cc tiu ti .

V D MU.

Bi mu 2.1.1. Tm im cc tr ca cc hm s sau

a.

2 2.

2

x xy

x b. 3 2(1 ) .y x x

c. 1

cos cos2 .2

y x x d.

2

1.

1

xy

x x

e. 2 42 1 .3y x x x

GII. a. TX: Ta c

( )

32

Bng bin thin:

2 4 0

1

Vy, hm s t cc i ti v hm s t cc tiu ti

b. TX: Ta c ( )( )

Bng bin thin:

3

5 1

0 0

108

3125

0

Vy, hm s t cc i ti 3 108

,5 3125

CDx y v hm s t cc tiu

ti

c. TX: Ta c v

[

[

Ta c ( ) v ( ) nn hm s t

cc i ti

Ta cng c

33

(

)

nn hm s t cc tiu ti

2

2 .3

kx

d. TX: Ta c

( )

( )

Bng bin thin:

1 0

2

Vy, hm s t cc i ti v

e. TX: Ta c

( ) | | {

( ) {

Bng bin thin:

0

s17

4

3

Vy, hm s t cc i ti 1 17

,2 4

CDx y v t cc tiu ti

34

BI TP TNG T.

Bi tp 2.1.1. Tm cc tr ca cc hm s sau

a. 3 22 2 1.y x x x b. cos sin .y x x

c. 4 24 5.y x x d. 22 .y x x x

35

VN II. CC TR CA HM S CHA THAM S

PHNG PHP.

Dng 1. Tm tham s m hm s t cc tr ti .

Nu hm s ( ) t cc tr ti im th ( ) hoc ti

khng c o hm. T y gii ra cc gi tr ca tham s .

Khi c cc gi tr , ta s dng mt trong hai cch sau y th :

+ Cch 1. Tnh , ri thay vo v s dng du hiu II,

xc nh ti gi tr no th hm t cc i hoc cc tiu. Cch ny

thng dng vi hm a thc, d dng ly .

+ Cch 2. Vi cc gi tr , ta suy ra v dng bng bin thin.

Cch ny thng dng khi cch 1 t ra phc tp trong vic ly .

Dng 2. Tm tham s m hm s c cc tr.

Ta thng lm nh sau:

+ Xc nh ( ).

+ Nu h s ca ( ) c cha th phi xt trng hp .

+ Khi , hm c cc tr khi i du. T y l lun ty theo

hm .

Dng 3. Tm tham s m hm s c cc tr tha mn iu kin no

.

Ta thng lm nh sau:

+ Xc nh ( ).

+ Nu h s ca ( ) c cha th phi xt trng hp .

+ Khi , hm f c cc tr khi i du. T y l lun ty theo

hm , gii ra cc khong gi tr ca . Vi trn, tm cc nghim ca

(chnh l cc cc tr) v s dng iu kin , tm ra cc gi tr .

36

Dng 4. ng thng i qua hai cc tr ca th hm s.

1. Hm s bc ba ( ) .

Chia ( ) cho ( ), ta c ( ) ( ) ( )

Khi , gi s ( ) ( ) l cc im cc tr th:

1 1 1

2 2 2

( ) ,

( ) .

y f x Ax B

y f x Ax B

Tc l cc im ( ) ( ) nm trn ng thng

.

2. Hm s phn thc

2( )( )

( )

P x ax bx cy f x

Q x dx e.

Gi s ( ) l im cc tr th 0

0

0

'( )

'( )

P xy

Q x.

Gi s hm s c cc i v cc tiu th phng trnh ng

thng i qua hai im cc tr y l:

'( ) 2

'( )

P x ax by

Q x d.

CH .

Hm s bc ba ( ) c cc tr

Phng trnh c hai nghim phn bit.

Khi nu l im cc tr th ta c th tnh gi tr cc tr

( ) bng hai cch:

+ ( )

+ , trong l phn d trong php

chia cho .

Ta tnh phn d bng cch ly chia cho v c

( )

V ( ) nn 0 0( )y x Ax B ; y l cc tr ca hm bc 3.

37

Khi s dng iu kin cn xt hm s c cc tr cn phi kim tra

li loi b nghim ngoi lai.

Khi gii cc bi tp loi ny thng ta cn s dng cc kin thc

khc na, nht l nh l Viette:

1 2

1 2

,

.

x

x

bx

a

cx

a

V D MU.

V d mu 1. Cho hm s 3 2 222 2

( ) (3 1) .3 3

f x x mx m x Tm

a. Hm s c cc tr.

b. Hm s t cc i ti .

c. Hm s c cc i, cc tiu l 1 2,x x v ( ) .

d. Hm s t cc i v cc tiu ti im c honh dng.

GII. Ta c ( ) ( )

a. Hm s c hai cc tr khi v ch khi phng trnh c hai

nghim phn bit 2 13

13m hoc

2 13.

13m (*)

b. Hm s t cc i ti khi ( ) hoc

2

3m (tha iu kin (*)).

Khi , ta c ( ) . V

bng bin thin ca , ta thy rng l im cc i ca hm s;

nh vy tha yu cu bi ton.

38

Khi 2

,3

m bng cch lm tng t, ta cng c 2

3m tha yu

cu bi ton.

Vy, vi hoc 2

3m th hm s t cc i ti .

c. Trc ht, hm s c hai cc tr khi tha ( ).

Lc , gi l hai cc tr ca hm s th l hai nghim ca

phng trnh v ta c .

Do , ( ) hoc

2

.3

m Kim tra iu kin, ta c 2

.3

m

V d mu 2. Chng minh rng vi mi hm s

( ) ( ) ( )

lun c cc i, cc tiu v khong cch gia chng l hng s.

GII. Ta c ( ) ( ) ( ) v

( ) ( )

Do , hm s cho lun c hai cc tr, vi mi gi tr ca .

Hai im cc tr ca hm s l v

T ( ) v ( )

.

Khong cch gia hai cc tr ca hm s l

| |

Vy, vi mi hm s cho lun c cc i, cc tiu v khong

cch gia chng l hng s.

V d mu 3. Cho hm s ( ) . Tm

th hm s cho c hai im cc tr nm v hai pha trc tung.

39

GII. Ta c ( ) Hm s c cc i v cc

tiu khi v ch khi iu ny ng vi mi .

Lc , hai im cc tr ca th hm s nm v hai pha trc tung

khi v ch khi hm s c hai im cc tr tri du, tc l

V d mu 4. Tm hm s c cc i, cc tiu

v hai im cc tr ca th hm s cng vi gc ta to thnh mt

tam gic c din tch bng 48.

GII. Ta c

th hm s c hai im cc tr kh v ch khi ( )

Cc im cc tr ca th l ( ) ( ).

Suy ra | | ( ) | |

Ta c tha mn ( ).

V d mu 5. Tm th hm s 3 21

( ) 3 12 2

mf x x x c hai cc

tr i xng nhau qua im ( )

GII. Trc ht, ta thy rng

( )

Nh vy, vi mi gi tr ca , th hm s lun c hai cc tr l

(

) (

)

Hai im cc tr ny i xng nhau qua im ( ) khi v ch khi l

trung im tc l

40

V d mu 6. Tm th hm s ( ) ( ) c

cc i, cc tiu to thnh mt tam gic vung.

GII. Ta c ( ) ( )

th hm s c ba im cc tr khi ( )

Cc im cc tr ca th l

( ) ( ) ( )

Suy ra ( ( ) ) ( ( ) )

V nn tam gic vung khi v ch khi

( ) ( )

Kt hp ( ) ta c gi tr cn tm l .

V d mu 7. Tm m th hm s

3 2 2 2( ) 3 3( 1) 3 1f x x x m x m

c cc i, cc tiu v cc im cc tr cch u gc ta O.

GII. Ta c ( ) Hm s cho c hai cc tr khi

v ch khi ( )

Lc , hai im cc tr ca th l

( ) ( )

Hai im cc tr ca th cch u gc ta khi v ch khi

( ( ))

41

BI TP TNG T.

Bi tp 2.2.1. Tm hm s 4 22( 2) 5y mx m x m c mt im

cc i 1

.2

x

Bi tp 2.2.2. Cho hm s 3 24 3y x mx x . Tm hm s c hai

im cc tr 1 2,x x tha 1 24 .x x

Bi tp 2.2.3. Cho hm s 3 21

(2 1) 33

y x mx m x c th l ( )

Xc nh ( ) c cc im cc i, cc tiu nm v cng mt pha

i vi trc tung.

Bi tp 2.2.4. Cho hm s 3 2(1 2 ) (2 ) 2y x m x m x m (1). Tm

cc gi tr ca th hm s (1) c im cc i, im cc tiu, ng

thi honh ca im cc tiu nh hn 1.

Bi tp 2.2.5. Cho hm s 3 2 2 33 3( 1)y x mx m x m m (1). Tm

hm s (1) c cc tr ng thi khong cch t im cc i ca th

hm s n gc ta bng 2 ln khong cch t im cc tiu ca

th hm s n gc ta .

Bi tp 2.2.6. Cho hm s 3 2 3 23( 1) 3 ( 2) 3 ( )my x m x m m x m m C .

Chng minh rng vi mi , th ( ) lun c 2 im cc tr v khong

cch gia 2 im cc tr l khng i.

Bi tp 2.2.7. Cho hm s 4 2 2( ) 2( 2) 5 5y f x x m x m m ( )mC .

Tm cc gi tr ca m th ( )mC ca hm s c cc im cc i, cc

tiu to thnh 1 tam gic vung cn.

43

CHUYN III. GI TR LN NHT V GI TR NH NHT CA HM S

1. nh ngha.

Gi s hm s xc nh trn min

a)

0 0

( ) , ,max ( )

: ( ) .D

f x M x DM f x

x D f x M

b)

0 0

( ) , ,min ( )

: ( ) .D

f x m x Dm f x

x D f x m

2. Tnh cht.

a) Nu hm s ng bin trn [ ] th

[ ; ][ ; ]

max ( ) ( ), min ( ) ( ).a ba b

f x f b f x f a

b) Nu hm s nghch bin trn [ ] th

[ ; ][ ; ]

max ( ) ( ), min ( ) ( )a ba b

f x f a f x f b .

VN : NG DNG O HM TM GTLN, GTNN

PHNG PHP.

1. Tm GTLN, GTNN ca hm s trn mt khong.

Tnh ( ).

Xt du ( ) v lp bng bin thin.

Da vo bng bin thin kt lun.

2. Tm GTLN, GTNN ca hm s lin tc trn mt on [ ]

Tnh ( )

Gii ptrnh ( ) tm c cc nghim 1 2, , , nx x x trn ;a b

44

(nu c).

Tnh 1 2, , , , , nf a f b f x f x f x .

So snh cc gi tr va tnh v kt lun.

1 2[ ; ]

max ( ) max ( ), ( ), ( ), ( ),..., ( ) ,na b

M f x f a f b f x f x f x

1 2[ ; ]min ( ) min ( ), ( ), ( ), ( ),..., ( ) .n

a bm f x f a f b f x f x f x

V D MU.

V d mu 1. Tm GTLN, GTNN ca cc hm s sau y:

a. 4 22 3y x x trn

1;1

2.

b. 3

22 43

3x

y x x trn on 4;0 .

c.

21

1

xy

x trn [-1;2].

d. 2cos2 4siny x x trn

0; .

2

e. 8 42sin cos 2y x x .

f. 3 23 72 124y x x x trn [-5;5].

GII. a. Tx Ta c ( )

Trn

1;1 ,

2 ta nhn cc gi tr

Ta c

1 41; 0 3; 1 2.

2 16y y y

45

Vy

11 ;1;122

1 41max ;min 1 2.

2 16y y y y

b. Tx Ta c

Ta c

( )

( ) ( )

( )

Vy,

4;0

max ( 3) (0) 4y y y v

4;0

16min ( 4) ( 1) .

3y y y

c. Tx Ta c

( ) [ ]

M

( ) ( ) ( )

Vy,

1;2

max (1) 2y y v

1;2

3min ( 1) (2) .

5y y y

d. Tx . t ( )

t Khi

02

x th

Xt hm s ( ) [ ]

Ta c

( )

[ ]

( ) ( ) (

)

Vy,

0;1

0;2

2max max ( ) 2 2

2y g t g v

0;1

0;2

min min ( ) (0) 2.y g t g

e. Tx Ta c ( ) ( )

46

t [ ]. Xt hm s ( ) ( ) [ ]

Ta c

( ) ( )

( ) ( ) (

)

Vy,

0;1

max max ( ) 1 3y g t g v

0;1

1 1min min ( ) .

3 27y g t g

f. Tx . Ta c

2 3 2

3 2

3 24 3 72 124 4,' 0

6,3 72 124

2xx x x x xy

xx x x

v trn [ ] khng xc nh ti .

Ta c ( ) ( ) ( )

Vy,

5;5

max 5 434y y v

5;5

min 2 0y y .

V d mu 2. Cho , 0, 1x y x y . Tm GTNN, GTLN ca biu thc

1 1

x yA

y x.

GII.

V Thay vo ta c

( )

( ) ( )

Ta c

( ) (

)

( )

Vy, GTLN ca l 1, xy ra khi hoc .

GTNN ca l 2

,3

xy ra khi 1

.2

x y

47

BI TP TNG T

Bi tp 3.0.1. Tm GTLN, GTNN ca cc hm s sau:

a) 3 22 3 12 1y x x x trn [1; 5].

b) 4 22 3y x x trn [3; 2].

c)

3 1

3

xy

x trn [0; 2] . d)

24 7 7

2

x xy

x trn [0; 2].

e) 2100y x trn [6; 8]. f) 2 4 .y x x

Bi tp 3.0.2. Tm GTNN v GTLN ca cc hm s sau:

a. 34

( ) 2sin sin3

f x x x trn

0;

2.

b. 2( ) sin2

xf x x trn

;

2 2.

c. 8 4( ) 2sin cos 2f x x x trn

;4 6

d. 3 23( ) 72 124x xf x x trn [-5;5].

e. 2( ) 4f x x x .

f. 6 2 3( ) 4(1 )f x x x trn [-1; 1].

49

CHUYN IV. NG TIM CN

PHNG PHP.

nh ngha.

ng thng 0x x gl ng tim cn ng ca th hm s

( )y f x nu t nht mt trong cc iu kin sau c tho mn

0

lim ( )x x

f x ;

0

lim ( )x x

f x ;

0

lim ( )x x

f x ;

0

lim ( ) .x x

f x

ng thng 0y y gl ng tim cn ngang ca th hm s

( )y f x nu t nht mt trong cc iu kin sau c tho mn

0lim ( )

xf x y ;

0lim ( )

xf x y

ng thng , 0y ax b a gl ng tim cn xin ca th

hm s ( )y f x nu t nht mt trong cc iu kin sau c tho mn

lim ( ) ( ) 0x

f x ax b ;

lim ( ) ( ) 0.x

f x ax b

CH .

a) Nu ( )

( )( )

P xy f x

Q x l hm s phn thc hu t.

Nu ( ) c nghim th th c tim cn ng 0x x .

Nu bc( ( )) bc( ( )) th th c tim cn ngang.

Nu bc( ( )) bc( ( )) th th c tim cn xin.

b) xc nh cc h s trong phng trnh ca tim cn xin, ta

c th p dng cc cng thc sau:

( )

lim ; lim ( )x x

f xa b f x ax

x

hoc

( )

lim ; lim ( ) .x x

f xa b f x ax

x

50

V D MU.

Bi mu 4.0.1. Tm tim cn ca cc th hm s sau

a.

3 2( )

2 1

xf x

x. b.

22

( )5 4

xf x

x x.

c.

2

1( )

1

xf x

x x. d. 2( ) 2 1 1f x x x x .

e. 3

2( )

1

xf x

x. f.

1( )

3

xf x

x.

GII. a. Ta c

( )

( )

b. Ta c

( )

( )

( )

c. Ta c

( )

( )

th hm s cho khng c TC.

d. Ta c

( )

( ( ) )

51

th hm s khng c TC.

e. Ta c

( )

( )

f. Ta c

( )

( )

Bi mu 4.0.2. Cho hm s

2( )

2

xf x

x( ). Gi l giao im hai tim

cn.

a. Trn ( ) tm cc im cch u hai ng tim cn ca ( ).

b. Tm thuc ( ) sao cho tng khong cch t M n hai tim cn

ca ( ) nh nht.

GII. th hm s cho c TC v TCN

Giao im hai tim cn ca th ( ) l ( )

a. Xt

00

0

.;2

2xM x

xC im cch u hai ng tim cn khi v

ch khi ( ) ( )

| | |

| | |

Lc ( ) v ( )

b. Xt

00

0

.;2

2xM x

xC Tng khong cch t n hai ng tim

cn ca ( ) l

52

( ) ( ) | | |

| | |

| |

| |

| |

Du bng xy ra khi v ch khi | |

Lc ( ) v ( )

BI TP TNG T.

Bi tp 4.0.1. Tm cc tim cn ca th cc hm s sau:

a)

10 3

1 2

xy

x b)

2 4 3.

1

x xy

x


a)

2

2

9

xy

x b)

2

2

2 3 3

1

x xy

x x

c)

4

3

4.

1

x xy

x


a) 2 4y x x b)

24 2

9

xy

x

c) 21

4 3y

x x d)

1

1

xy x

x

e) 3 2 33y x x f)

2 3 2

2

x xy

x.

53

CHUYN V. KHO ST S BIN THIN V V TH HM S

PHNG PHP.

1. Cc bc kho st s bin thin v v th ca hm s.

Tm tp xc nh ca hm s.

Xt s bin thin ca hm s:

+ Tnh .

+ Tm cc im ti o hm bng 0 hoc khng xc nh.

+ Tm cc gii hn ti v cc, gii hn v cc v tm tim cn (nu

c).

+ Lp bng bin thin ghi r du ca o hm, chiu bin thin,

cc tr hm s.

V th ca hm s:

+ Tm im un ca th (i vi hm s bc ba v hm s trng

phng).

Tnh .

Tm cc im ti v xt du

+ V cc ng tim cn (nu c) ca th.

+ Xc nh mt s im c bit ca th nh giao im ca

th vi cc trc to (trong trng hp th khng ct cc trc to

hoc vic tm to giao im phc tp th c th b qua). C th tm

thm mt s im thuc th c th v chnh xc hn.

+ Nhn xt: Ch ra trc i xng, tm i xng (nu c) ca th.

2. Hm s bc ba 3 2 ( 0)y ax bx cx d a :

Tp xc nh .

th lun c mt im un v nhn im un lm tm i xng.

54

3. Hm s trng phng 4 2 ( 0)y ax bx c a :

Tp xc nh .

th lun nhn trc tung lm trc i xng.

4. Hm s nht bin

( 0, 0)ax b

y c ad bccx d

:

Tp xc nh \ dD c .

th c mt TC l d

xc

v mt TCN l a

yc

.

Giao im ca hai tim cn l tm i xng ca th hm s.

Hm s khng c cc tr.

5. Hm s hu t

2

' '

ax bx cy

a x b:

Tp xc nh '\ 'b

Da

.

th c mt tim cn ng l '

'

bx

a v mt tim cn xin.

Giao im ca hai tim cn l tm i xng ca th hm s.

V D MU.

Bi mu 5.0.1. Kho st s bin thin v v th ca cc hm s

a. 3 2( ) 3 4f x x x . b. 3 2( ) 3 4 2f x x x x .

c. 4 2( ) 2 2f x x x . d. 4

2 3( )2 2

xf x x .

e.

2( )

2 1

xf x

x f.

1( )

1

xf x

x

g.

2 3 6( )

1

x xf x

x h.

2 2 1( ) .

1

x xf x

x

55

GII. a.

b.

c.

d.

e.

f.

x

y

1 x

y

1

x

y

1

x

y

1

x

y

1 x

y

1

56

g.

h.

BI TP TNG T.

Bi tp 5.0.1. Kho st s bin thin v v th ca cc hm s

a) 3 23 9 1y x x x b) 3 23 3 5y x x x

c) 3 23 2y x x d) 2( 1) (4 )y x x

e) 3

2 1

3 3

xy x f) 3 23 4 2.y x x x


a) 4 22 1y x x b) 4 24 1y x x

c) 4

2 532 2

xy x

d) 2 2( 1) ( 1)y x x

e) 4 22 2y x x f) 4 22 4 8.y x x


a)

1

2

xy

x b)

2 1

1

xy

x

c)

3

4

xy

x d)

1 2

1 2

xy

x

x

y

1 x

y

1

57

CHUYN VI. MT S BI TON LIN QUAN KHO ST HM S

VN I. S TNG GIAO CA CC TH

PHNG PHP.

1. Cho hai th ( ) ( ) v ( ) ( ). tm honh

giao im ca ( ) v ( ) ta gii phng trnh ( ) ( ) ( ) (gi l

phng trnh honh giao im). S nghim ca phng trnh ( )

bng s giao im ca hai th.

2. th hm s 3 2 ( 0)y ax bx cx d a ct trc honh ti 3 im

phn bit Phng trnh 3 2 0ax bx cx d c 3 nghim phn bit

Hm s 3 2y ax bx cx d c cc i, cc tiu v . 0CD CTy y .

V D MU.

Bi mu 6.1.1. Cho hm s 3 23 2y x m x m c th ( ). Tm

th ( ) ct trc honh ti ng hai im phn bit.

GII. ( ) ct trc honh ti ng hai im phn bit th trc ht ( )

phi c 2 im cc tr, tc l 0y c 2 nghim phn bit

2 23 3 0x m c 2 nghim phn bit .

Khi ' 0y x m .

( ) ct ti ng 2 im phn bit hoc .

Ta c + 3( ) 0 2 2 0 0y m m m m (loi).

+ 3( ) 0 2 2 0 0 1.y m m m m m

58

Vy, th ( ) ct trc honh ti ng hai im phn bit khi

1.m

Bi mu 6.1.2. Cho hm s 4 2 1y x mx m c th l mC . nh

th mC ct trc honh ti bn im phn bit.

GII. Phng trnh honh giao im ca ( ) vi trc honh

4 2 1 0.x mx m (1)

t 2 , 0t x t . Khi : (1) 2 1 0t mt m (2)

1,

1.

t

t m

YCBT (1) c 4 nghim phn bit (2) c 2 nghim dng phn

bit 0 1 1m

1,

2.

m

m


2 1

2

xy

x c th l ( ) Chng minh rng

ng thng y x m lun ct th ( ) ti hai im phn bit

Tm on c di ngn nht.

GII. Phng trnh honh giao im ca ( ) v

2 1

2

xx m

x

2

2,

( ) (4 ) 1 2 0. (1)

x

f x x m x m

Do (1) c vi mi v

2( 2) ( 2) (4 ).( 2) 1 2 3 0,f m m m

nn ng thng lun lun ct th ( ) ti hai im phn bit

Ta c ;A A B By m x y m x nn

59

2 2 2 2( ) ( ) 2( 12).B A B AAB x x y y m

Suy ra AB ngn nht 2AB nh nht 0m . Khi 24AB .

BI TP TNG T.

Bi tp 6.1.1. Cho hm s 3 26 9 6y x x x c th l ( ) nh

ng thng ( ): 2 4d y mx m ct th ( ) ti ba im phn bit.

Bi tp 6.1.2. Cho hm s 3 21 2

3 3y x mx x m co o thi ( )mC . T m

e ( )mC ca t trc hoa nh ta i 3 ie m pha n bie t co to ng b nh phng ca c

hoa nh o l n hn 1 .

Bi tp 6.1.3. Cho hm s 3 3 2y x x . Vit phng trnh ng thng

ct th ( ) ti 3 im phn bit sao cho 2Ax v 2 2BC .

Bi tp 6.1.4. Cho hm s 4 2(3 2) 3y x m x m c th l ( ) Tm

ng thng ct th ( ) ti 4 im phn bit u c

honh nh hn 2.


2 2

1

xy

x( ) Tm ng thng ( ):

2y x m ct ( ) ti hai im phn bit sao cho 5AB .


2 1

1

xy

x( ) Tm ng thng

ct ( ) ti hai im phn bit sao cho vung ti

.

61

VN II. BIN LUN NGHIM PHNG TRNH

PHNG PHP.

1. Bin lun nghim phng trnh da vo th hm s va kho

st.

BI TON. Bin lun bng th s nghim ca phng trnh

( )f x m ( )

Nhn thy (*) l phng trnh honh giao im ca th hai hm

s: ( )y f x v y m .

S nghim ca phng trnh l s giao im ca th hai hm s

trn.

Da vo th, a ra kt lun.

2. Bin lun nghim phng trnh c cha gi tr tuyt i.

a. Cc php bin i th thng gp.

Gi s hm s ( ) c th ( ).

+ ( )y f x : c th i xng vi ( ) qua .

+ ( )y f x : c th i xng vi ( ) qua .

+ ( )y f x : c th i xng vi ( ) qua gc ta .

+ ( )y f x b : tnh tin ( ) theo Oy b n v.

+ ( )y f x a : tnh tin ( ) theo Ox a n v.

+ ( )x f y : i xng vi ( ) qua ng phn gic gc phn t th

nht .

+ ( )y f x : Ta c

( ) khi ( ) 0,( )

( ) khi ( ) 0.

f x f xy f x

f x f x

Suy ra th hm s ( )y f x gm hai phn:

62

Phn 1: th hm s ( )y f x ng vi 0y .

Phn 2: i xng phn cn li qua Ox.

+ y f x : Ta c f x f x nn hm s y f x l hm s

chn, do th hm s y f x i xng nhau qua .

Suy ra th hm s ( )y f x gm hai phn:

Phn 1: th hm s ( )y f x ng vi 0x .

Phn 2: i xng phn 1 qua .

+ y f x : v th hm s ( )y g x f x , sau v th

hm s ( )y g x .

+ ( )y f x : Nu ;o ox y thuc th hm s ny th ;o ox y

cng thuc th hm s, do th hm s nhn trc lm trc i

xng.

Suy ra th hm s gm hai phn:

Phn 1: th hm s ( ) ng vi 0y .

Phn 1: i xng phn 1 qua .

b. Bin lun nghim phng trnh.

T th hm s kho st, v th hm s cha gi tr tuyt i,

bng cch da vo cc php bin i trn.

Tin hnh bin lun nh trong phn 1.

V D MU.

Bi mu 6.2.1. Bin lun theo s nghim ca cc phng trnh sau:

a. 3 23 4x x m . b. 4 2 22 0x x m .

c.

2 12

1

xm

x d. 3 23 2xx m .

63

GII. a. Phng trnh cho l phng trnh honh giao im ca th

hai hm s v ng thng

S nghim ca phng trnh cho l s giao im ca th hai hm

s trn.

th hm s l ng cong ( ), th hm s

l ng thng song song hoc trng vi trc honh.

Quan st th, ta thy rng

+ hoc : phng

trnh cho c mt nghim duy nht.

+ hoc phng

trnh cho c hai nghim phn bit

(mt nghim n v mt nghim kp).

+ phng trnh

cho c ba nghim phn bit.

b. Phng trnh cho l phng trnh honh giao im ca th

hai hm s v ng thng

S nghim ca phng trnh cho

l s giao im ca th hai hm s

trn.

th hm s l

ng cong ( ), th hm s l

ng thng song song hoc trng vi

trc honh.


+ phng trnh cho c hai nghim phn bit.

phng trnh cho c ba nghim.

64

c. t

2 1( ) .

1

xf x

x Ta thy rng

| |

| | (| |) {

( )

( )

Do , th hm s (| |) gm hai phn:

+ Phn 1: th hm s ( ) bn phi trc tung.

+ Phn 2: i xng phn 1 qua trc tung.

Phng trnh cho l phng trnh

honh giao im ca th hai hm s

(| |) v ng thng

S nghim ca phng trnh cho l

s giao im ca th hai hm s trn.

th hm s (| |) l ng cong

( ), th hm s l ng thng

song song hoc trng vi trc honh.


+

phng trnh

cho v nghim.

+

phng trnh cho c hai

nghim phn bit.

+

phng trnh cho c mt nghim duy

nht.

d. t ( ) . Ta c

| | | ( )| { ( ) ( )

( ) ( )

Nh vy, th hm s | ( )| gm hai phn:

+ Phn 1: th hm s ( ), pha trn trc .

65

+ Phn 2: i xng qua trc phn th hm s ( ) nm

di .

Phng trnh cho l phng trnh honh giao im ca th

hai hm s | ( )| v ng thng

S nghim ca phng trnh cho l s giao im ca th hai hm

s trn.

th hm s | ( )| l ng

cong ( ), th hm s l

ng thng song song hoc trng vi

trc honh.

Quan st th, ta thy rng:

+ phng trnh cho

c hai nghim phn bit.

+ phng trnh cho

c bn nghim phn bit.

+ phng trnh cho c su nghim phn bit.

+ phng trnh cho c ba nghim phn bit.

+ phng trnh cho v nghim.

Bi mu 6.2.2. Tm phng trnh c ba

nghim phn bit, trong c ng hai nghim ln hn .

GII. Phng trnh cho l phng trnh honh giao im ca th

hai hm s v ng thng

Qua th ta thy, phng trnh c ba nghim phn bit trong c

ng hai nghim ln hn khi v ch khi

66

Bi mu 6.2.3. Cho hm s Tm phng trnh

3 2 3 23 3x x m m c ba nghim phn bit.

GII.

PT 3 2 3 23 3x x m m 3 2 3 23 1 3 1x x m m .

t 3 23 1k m m .

S nghim ca phng trnh cho bng s giao im ca th ( )

vi ng thng

Da vo th ( ) ta c, phng trnh cho c 3 nghim phn bit

khi v ch khi 1 5k hay ( 1;3)\{0;2}.m

BI TP TNG T.

Bi tp 6.2.1. Kho st s bin thin v v th ( ) ca hm s. Dng

th ( ) bin lun theo m s nghim ca phng trnh:

a) 3 33 1; 3 1 0.y x x x x m

b) 4

2 4 22 2; 4 4 2 0.2

xy x x x m

c) 3 2 3 23 6;cos 3cos 6 0.y x x x x m

Bi tp 6.2.2. Kho st s bin thin v v th ( ) ca hm s. Dng

th ( ) bin lun theo m s nghim ca phng trnh:

a) 3 2 3 2 3 2( ): 3 6; ( ): 3 6 ; 3 6 3 0.C y x x T y x x x x m

b) 2 2 2( ): ( 1) (2 ); ( ): ( 1) 2 ;( 1) 2 .C y x x T y x x x x m

c)

1 1( ): ; ( ): ; 1 1 0

1 1

x xC y T y x m x

x x

67

VN III. BI TON TIP XC GIA CC NG

PHNG PHP.

Bi ton 1: Vit PTTT ca ( ) ( ) ti im 0 0 0;M x y :

Nu cho th tm ( ).

Nu cho th tm l nghim ca phng trnh ( ) .

Tnh ( ). Suy ra ( ) ( ).

Phng trnh tip tuyn l ( )( ) ( ).

Bi ton 2: Vit PTTT ca ( ) ( ) bit c h s gc cho

trc.

Cch 1: Tm to tip im.

Gi ( ) l tip im. Tnh ( )

c h s gc suy ra ( ) (1)

Gii phng trnh (1), tm c v tnh ( ). T vit

phng trnh ca .

Cch 2: Dng iu kin tip xc.

Phng trnh ng thng c dng

tip xc vi ( ) khi v ch khi h phng trnh sau c nghim:

( )

'( )

f x kx m

f x k ( )

Gii h ( ), tm c . T vit phng trnh ca .

CH . H s gc ca tip tuyn c th c cho gin tip nh sau:

+ to vi chiu dng trc honh gc th

+ song song vi ng thng th

+ vung gc vi ng thng th 1

.ka

+ to vi ng thng mt gc th

68

tan .1

k a

ka

Bi ton 3: Vit PTTT ca ( ) ( ) bit i qua im ( ; )A AA x y .

Cch 1: Tm to tip im.

Gi ( ) l tip im. Khi ( ) ( ).

Phng trnh tip tuyn ti : ( ) ( ) ( )

i qua ( ; )A AA x y nn: ( ) ( ) ( ) (2)

Gii phng trnh (2), tm c .

T vit phng trnh ca .

Cch 2: Dng iu kin tip xc.

Phng trnh ng thng i qua ( ; )A AA x y v c h s gc :

( )

tip xc vi (C) khi v ch khi h phng trnh sau c nghim

( ) ( )

'( )

A Af x k x x y

f x k( )

Gii h ( ), tm c (suy ra ). T vit phng trnh tip

tuyn .

Bi ton 4. Tm nhng im trn ng thng m t c th v

c 1, 2, 3, tip tuyn vi th ( ) ( ).

Gi s ( ) .

Phng trnh ng thng qua c h s gc :

( )

tip xc vi ( ) khi h sau c nghim:

( ) ( ) (1),

'( ) (2).

M Mf x k x x y

f x k

69

Thay t (2) vo (1) ta c ( ) ( ) ( ) (3)

S tip tuyn ca ( ) v t bng s nghim ca (3).

V D MU.


2( )

1

xy f x

x (C).

a. Vit PTTT ca ( ) ti im c honh bng 2.

b. Vit PTTT ca ( ) ti giao im ca ( ) vi trc honh.

c. Vit PTTT ca ( ) ti giao im ca ( ) vi trc tung.

d. Vit PTTT ca ( ) bit tip tuyn song song vi 1

100.3

y x

e. Vit PTTT ca ( ) bit tt vung gc vi .

GII. Hm s xc nh trn Ta c

( )

( )

a. Ti im c honh bng , ta c ( ) ( )

Phng trnh tip tuyn cn tm l

b. th ( ) ct trc honh ti im ( ) Lc

( )


c. Tip tuyn song song vi ng thng khi v ch khi

( )

Vi ta c ( ) v phng trnh tip tuyn l

70

Vi ta c ( ) v phng trnh tip tuyn l

d. ng thng c h s gc

Tip tuyn vung gc khi v ch khi ( ) ( ) hay

( ) ( )


Bi mu 6.3.2. Cho hm s 3 2( ) 3 2y f x x x c th ( ).

a. Vit PTTT qua

23; 2

9A n ( )

b. Tm tip tuyn vi ( ) c h s gc nh nht.

GII. Hm s xc nh trn v ( )

a. ng thng i qua , h s gc c phng trnh

(

)

v ( ) tip xc nhau khi v ch khi h sau c nghim:

{ (

)

Trong h trn, thay trong phng trnh th nht bi trong

phng trnh th hai, gii c

Phng trnh tip tuyn ti cc im c honh tng ng l

71

b. Ta c ( ) ( ) du bng xy ra khi

v ch khi . Nh vy, trong cc tip tuyn ca ( ) tip tuyn c h s

gc nh nht bng . Lc tip im c ta ( ) v phng trnh

tip tuyn l

Bi mu 6.3.3. Vit phng trnh tip tuyn ca th hm s

( ) bit tip tuyn ct hai trc ta ln lt

ti sao cho .

GII. V tip tuyn ct ln lt ti vi nn

( )

Vi ( ) , phng trnh v nghim.

Vi ( ) Lc ta c hai

tip tuyn vi phng trnh ln lt l


1

2 1

xy

x. Chng minh rng vi mi ,

ng thng lun ct ( ) ti 2 im phn bit . Gi

1 2,k k ln lt l h s gc ca cc tip tuyn vi ( ) ti v Tm

tng 1 2k k t gi tr ln nht.

GII. PT honh giao im ca v ( )

1

2 1

xx m

x

2

1

2

( ) 2 2 1 0 (*)

x

g x x mx m

72

V 2 2 2 0,g m m m v

10

2g nn (*) lun c 2 nghim

phn bit 1 2,x x . Gi s: 1 1 2 2( ; ), ( ; )A x y B x y .

Theo nh l Viet ta c

1 2 1 2

1;

2

mx x m x x .

Tip tuyn ti A v B c h s gc l

1 22 21 2

1 1; .

(2 1) (2 1)k k

x x

T 21 2 4( 1) 2 2k k m . Du xy ra 1m .

Vy: 1 2k k t GTLN bng khi .


2

2 3

xy

x(1). Vit phng trnh tip tuyn

ca th hm s (1), bit tip tuyn ct trc honh, trc tung ln lt

ti hai im phn bit v tam gic cn ti gc ta .

GII.

Gi 0 0( ; )x y l to ca tip im. Suy ra

0 2

0

1( ) 0.

(2 3)y x

x

cn ti nn tip tuyn song song vi ng thng y x

(v tip tuyn c h s gc m).

Ngha l

0 2

0

1( ) 1

(2 3)y x

x, suy ra hoc

+ Vi 0 01; 1x y : 1 ( 1)y x y x (loi).

+ Vi 0 02; 0x y : 0 ( 2) 2y x y x (nhn).

Vy phng trnh tip tuyn cn tm l: 2y x .

73

BI TP TNG T.

Bi tp 6.3.1. Vit phng trnh tip tuyn ca (C) ti im c ch ra:

a) ( ) 3 23 7 1y x x x ti ( ).

b) ( ) 4 22 1y x x ti ( )

c) ( )

3 4

2 3

xy

x ti ( )

Bi tp 6.3.2. Vit phng trnh tip tuyn ca ( ) ti im c ch ra:

a) ( ) 3 3 1y x x ti im c honh l nghim ca

b) ( )

3( 2)

1

xy

x ti im c

c) ( )

1

2

xy

x ti cc giao im ca ( ) vi trc honh, trc tung.


3

1

xy

x. Cho im ( ; )o o oM x y thuc th

( ) Tip tuyn ca ( ) ti ct cc tim cn ca ( ) ti cc im v

. Chng minh l trung im ca on thng .

Bi tp 6.3.4. Cho hm s y =

2 1

1

x

x. Lp phng trnh tip tuyn ca

th ( ) sao cho tip tuyn ny ct cc trc ln lt ti cc im

v tho mn .

Bi tp 6.3.5. Vit phng trnh tip tuyn ca ( ) 4 21 3

32 2

y x x

bit song song vi ng thng .

Bi tp 6.3.6. Vit PTTT ca ( ) 3

22 3 13

xy x x bit vung gc

vi ng thng 28

xy .

Bi tp 6.3.7. Vit phng trnh tip tuyn ca ( ) 3 3 2y x x ;

bit i qua im A(2; 4).

75

VN IV. MT S BI TON LIN QUAN IM C

BIT TRN TH HM S

PHNG PHP.

Bi ton 1: Tm cc im trn th hm s hu t ( )

( )

P xy

Q x c to

l nhng s nguyn.

+ Phn tch ( )

( )

P xy

Q x thnh dng ( )

( )

ay A x

Q x, vi ( ) l a

thc, l s nguyn.

+ Khi

x

y ( ) l c s ca . T ta tm cc gi tr

nguyn ( ) l c s ca .

+ Th li cc gi tr tm c v kt lun.

Bi ton 2: Tm nhng im trn th hm s c tng khong cch

n hai trc ta nh nht .

+ Gi ( ) vi ( ) v tng khong cch t n hai trc

l , ta c | | | |.

+ Xt cc khong cch t n hai trc khi nm cc v tr c

bit: trn trc honh, trn trc tung .

+ Sau xt tng qut, nhng im M c honh , hoc tung

ln hn honh hoc tung ca khi nm trn hai trc , suy ra

cch tm GTLN-GTNN ca .

Bi ton 3: Tm nhng im c nh m h th hm s lun i qua.

Cho h ng cong ( ) ( ) ( l tham s).

im ( ) cho trc bt k nm trn ( ) khi v ch khi

76

( ) ( )

Xem ( ) l phng trnh theo n .

Tu theo s nghim ca ( ) ta suy ra s th ca h ( ) i qua

+ Nu ( ) nghim ng vi mi th mi th ca h ( ) u

i qua . Lc ny, c gi l im c nh ca h ( ).

+ Nu ( ) c nghim phn bit th c th ca h ( ) i

qua .

+ Nu ( ) v nghim th khng c th no ca h ( ) i qua

.

CH .

+ Phng trnh :

0, 0,

0.

AAm B m

B

v nghim

0,

0.

A

B

+ Phng trnh 2 0:Am Bm C

2 0 0, 0, 0.Am Bm C A B C

2 0Am Bm C v nghim

2

0,

0,

0,

4 0.

A B

C

A

B AC

Bi ton 4. Tm qu tch ca mt vi im c bit trn h th hm

s.

NHN XT. Tm tp hp im ( ) trong mt phng to l tm

phng trnh ca tp hp im .

Cc bc sau y thng c s dng vi bi ton m ta d

dng tnh c to ca im M.

77

1. Tm iu kin (nu c) ca tham s tn ti im

2. Tnh to im theo tham s .

C cc trng hp xy ra:

Trng hp 1.

( ),:

( ).

x f mM

y g m

Kh tham s gia v , ta c mt h thc gia c lp vi

c dng ( ) (gi l phng trnh qu tch).

Trng hp 2.

( ),:

( ).

x a constM

y g m

Khi im M nm trn ng thng .

Trng hp 3.

( ),:

( ).

x f mM

y b const

Khi im nm trn ng thng .

3. Gii hn qu tch: Da vo iu kin (nu c) ca ( bc 1),

ta tm c iu kin ca hoc tn ti im ( ) l gii hn

ca qu tch.

4. Kt lun: Tp hp cc im c phng trnh ( )

(hoc , hoc ) vi iu kin ca hoc ( bc 3).

Trong trng hp ta khng th tnh c to ca im M theo

tham s m m ch thit lp c mt h thc cha to ca M th ta

tm cch kh tham s m trong h thc tm c h thc dng

( )

V D MU.

Bi mu 6.4.1. Tm trn th hm s

3( 1)

2

xy

x cc im c ta l

s nguyn.

78

GII. Ta vit li

iu kin cn v im thuc ( ) c ta nguyn

l nhn cc gi tr . Hay nhn cc gi tr

T ta c 6 im trn th hm s cho c ta nguyn l

( ) ( ) ( ) ( ) ( ) ( )

Bi mu 6.4.2. (A 2014) Tm trn th hm s

2

1

xy

x cc im

sao cho khong cch t n ng thng bng

GII.

Gi thuc th hm s cho. Lc

00 0

0

2; 1.,

1

xM x x

x

Khong cch t n ng thng l

|

|

Ta c [

[

Suy ra ta im cn tm l ( ) ( )

Bi mu 6.4.1. Cho hm s 4 2( 1)y x m x m . Tm im c nh ca

h th hm s cho khi m thay i.

GII. im ( ) l im c nh ca h th hm s khi v ch khi

( )

79

2 4 20 0 0 0( 1) ,x m y x x m

20

4 20 0 0

1 0

0

x

y x x

0 0

0 0

1 1,

0 0.

x x

y y

Vy, h th hm s cho c hai im c nh l ( )


2

2 1

xy

x c th ( ). Chng minh rng

ng thng lun i qua 1 im c nh ca ng

cong ( ) khi bin thin.

GII. Phng trnh honh giao im ca (C) v (d)

21

2 1

xmx m

x

2 ( 1)(2 1)x mx m x , iu kin: 1

2x ;

2 2 1 ( 1)(2 1)x x m x x

( 1)(2 1) 3 3m x x x

Giao im c nh ca ( ) v ( ) c honh l nghim ca h

phng trnh:

( 1)(2 1) 0

3 3 0

x x

x

11

2

1

x x

x

1x 1.y

Vy, vi mi gi tr ca , ng thng lun ct th ( ) ti mt

im c nh l ( )

Bi mu 6.4.5. Tm qu tch cc im cc i ca th hm s

( ) ( ) ( )

80

GII. Ta c [ ( ) ( )] v

Bng cch lp BBT ca hm , ta nhn thy rng hm s lun c cc

i v cc tiu ti , trong im cc i ca th

hm s l ( ( ))

Th bng cch thay vo hm , ta c .

Vy, qu tch cc im cc i ca th hm s l ng cong c

phng trnh .

Bi mu 6.4.6. Cho hm s .

a. Vit phng trnh tip tuyn ti cc im c nh m th hm s

lun i qua vi mi gi tr ca Tm qu tch giao im ca cc tip

tuyn khi thay i.

b. Hy xc nh tt c cc gi tr ca im cc i v im cc tiu

ca ( ) v hai pha khc nhau ca ng trn (pha trong v pha

ngoi)

GII. a. Ta c ( )

Do th lun i qua hai im c nh ( ) v ( ).

Ta c .

Tip tuyn ti c phng trnh ( )( ).

Tip tuyn ti c phng trnh ( )( ) .

Giao im ca hai tip tuyn c ta tha mn h

{ ( )( )

( )( ) {

( ) ( )

81

T h trn, suy ra

( )( ) ( )( )

Vy, qu tch cn tm l ng cong c phng trnh

b. ng trn cho c phng trnh ( ) ( ) ( )

ng trn ( ) c tm ( ) v bn knh .

( ) c im cc i ( ) v im cc tiu ( ).

Ta c ( ) ( ) ( )

Suy ra im nm ngoi ng trn ( ) Do yu cu bi ton xy

ra khi v ch khi im nm trong ng trn ( ), tc l

( )

BI TP TNG T.

Bi tp 6.4.1. Tm cc im trn th ( )

2

1

xy

x c to nguyn.

Bi tp 6.4.2. Tm trn th ( ) hai im i xng nhau qua

Bi tp 6.4.3. Tm trn th ( ) hai im i

xng nhau qua im ( )

Bi tp 6.4.4. Tm cc im thuc th

2( ):

2

xC y

x sao cho

( ) ( )

Bi tp 6.4.5. Tm cc im c nh ca h th ( ) c phng trnh

sau:

82

a) 3 2( 1) 2 ( 2) 2 1.y m x mx m x m

c) 4 22 4 1.y mx x m

c)

( 1) 2( 1, 2).

m xy m m

x m

Bi tp 6.4.6. Chng minh rng h th ( ) ( )

( ) c ba im c nh thng hng. Vit phng

trnh ng thng i qua ba im c nh .

Bi tp 6.4.7. Cho ( ) v ( )

1) Tm ( ) v ( ) ct nhau ti hai im phn bit .

2) Tm tp hp cc trung im ca on thng ; trong :

a) ( )

1

1

xy

x v ( )

b) ( )

2( 2)

1

xy

x v ( ) l ng thng i qua ( ) v c h

s gc .

83

PHN 2. MT S VN

NNG CAO

85

CHUYN I. S BIN THIN - CC TR V GI TR NH NHT GI

TR LN NHT CA HM S V D.

V d 1. Tm m hm s

( ) ( ) ( )

ng bin trn tp hp cc gi tr sao cho | | .

GII. Trc ht, ta c | | hoc ; v

( ) ( ) ( )

D dng thy rng ( )

Ta c bng bin thin ca :

+ 0 - 0 +

Hm s cho tng trn [ ] [ ] khi v ch khi [ ] [ ]

nm trong min nghim ca bt phng trnh ( ) , tc l

[

{

[

V d 2. Tm hm s c cc i, cc tiu v C. CT . Vit phng trnh ng thng i qua hai im cc tr ca th hm s.

GII. Hm s c cc i v cc tiu khi v ch khi c hai

nghim phn bit, hay .

86

Ly chia cho ta c

Gi l hai im cc tr ca hm s. Lc , l hai nghim

ca , tc l ( ) ( ) . Do

( )

( )

( )

( )

T

(

) (

)

( ) ( )

Theo nh l Viette, ta c

{

Thay vo ( ) ta c (tha iu kin c cc tr)

V d 3. Tm hm s 3 2( ) 3 1f x x mx c cc tr v tm qu tch

cc im cc i ca th hm s khi thay i.

GII. Ta c ( )

Hm s c (hai) cc tr khi v ch khi .

Khi th l im cc i ca hm s.

Khi th l im cc i ca hm s v

T ta c 2

xm thay vo , ta c

3

1.2

xy

Vy, qu tch cc im cc i l th hm s

87

V d 4. Tm hm s c cc i, cc tiu v

C. CT . Vit phng trnh ng thng i qua hai im cc tr ca

th hm s.

GII. Hm s c cc i v cc tiu khi v ch khi c hai

nghim phn bit, hay .

Ly chia cho ta c

Gi l hai im cc tr ca hm s. Lc , l hai nghim

ca , tc l ( ) ( ) . Do

( )

( )

( )

( )

T

(

) (

)

( ) ( )

Theo nh l Viette, ta c

{

Thay vo ( ) ta c (tha iu kin c cc tr)

88

V d 5. Tm hm s 3 2( ) 3 1f x x mx c cc tr v tm qu tch

cc im cc i ca th hm s khi thay i.

GII. Ta c ( )

Hm s c (hai) cc tr khi v ch khi .

Khi th l im cc i ca hm s.

Khi th l im cc i ca hm s v

T ta c 2

xm thay vo , ta c

3

1.2

xy

Vy, qu tch cc im cc i l th hm s

V d 6. Tm GTNN v GTLN ca

2

2

)2(6

2 2 1

xy xS

xy y vi 2 2 1x y .

GII. Nu th , lc

Nu th t ta c

( ) [

Bng bin thin ca

0 0

2 3 2

T bng bin thin, ta c:

GTLN ca l 3, xy ra khi v , tc l

89

GTNN ca l xy ra khi v tc l

V d 7. Cho ba s thc [ ]. Tm GTNN ca biu thc

GII. Xt hm s

( )

[ ]

( )

Suy ra hm ng bin trn [ ] nn

( ) ( )

( ) [ ]

( )

Suy ra hm nghch bin trn [ ], do vy

( ) ( )

( ) [ ]

( )

Do hm nghch bin trn [ ] v suy ra

( ) ( )

Vy GTNN ca l 7, xy ra khi

90

BI TP.

Bi tp 1*. Cho hm s 3 22x 3(2 1) 6 ( 1) 1y m x m m x c th

( ) Tm hm s ng bin trn khong (2; ).

Bi tp 2*. Cho hm s 2 3 21

( 1) ( 1) 2 13

y m x m x x . Tm

hm nghch bin trn khong ( ;2)K .

Bi tp 3*. Cho hm s 3 22 3 1y x mx (1). Tm cc gi tr ca m

hm s (1) ng bin trong khong 1 2( ; )x x vi 2 1 1x x .

Bi tp 4*. Tm hm s 3 2 22 1

( ) ( 1) ( 4 3)3 2

f x x m x m m x

c cc i, cc tiu. Gi 1 2;x x l hai cc tr ca hm s, tm gi tr ln nht

ca 1 2 1 22( )A xx xx .

Bi tp 5*. Cho hm s 3 21

13

y x mx mx . Xc nh m hm s

cho t cc tr ti 1 2,x x sao cho 1 2 8x x .

Bi tp 6*. Cho hm s 3 2 2 3 23 3(1 )y x mx m x m m . Vit phng

trnh ng thng qua hai im cc tr ca th hm s.

Bi tp 7*. Cho hm s 3 23 2y x x mx c th l ( ). Xc nh

( ) c cc im cc i v cc tiu cch u ng thng

Bi tp 8*. Cho hm s 3 2( ) 2 3( 3) 11 3y f x x m x m( mC ). T m m

( )mC c hai im cc tri 1 2,M M sao cho cc im 1 2,M M v ( )

thng hng.

Bi tp 9*. Cho hm s 3 23 2y x x mx m . Xc nh th

hm s c cc im cc i v cc tiu nm v hai pha i vi trc

honh.

91

Bi tp 10*. Cho hm s 3 2 2(2 1) ( 3 2) 4y x m x m m x ( l

tham s) c th l ( ). Xc nh ( ) c cc im cc i v cc

tiu nm v hai pha ca trc tung.

Bi tp 11*. Cho hm s 3 23( 1) 9 2y x m x x m (1) c th l

( ) Vi gi tr no ca th th hm s c im cc i v im cc

tiu i xng vi nhau qua ng thng 1

2y x .

Bi tp 12*. Cho hm s 3 23 2y x x mx c th l ( ) Tm

( ) c cc im cc i, cc tiu v ng thng i qua cc im cc tr

song song vi ng thng 4 3y x .

Bi tp 13*. Cho hm s 3 2 7x 3y x mx c th l ( ). Tm


vung gc vi ng thng 3 7y x .

Bi tp 14*. Cho hm s 3 23 2y x x mx c th l ( ). Tm


to vi ng thng 4 5 0x y mt gc 045 .

Bi tp 15*. Cho hm s 2 2 32 3( 1) 6y x m x mx m (1). Tm

th ca hm s (1) c hai im cc tr sao cho tam gic vung

ti , vi (4;0)C .

Bi tp 16*. Cho hm s ( ) ( ) Vi

nhng gi tr no ca th th ( ) c im cc i v im cc tiu,

ng thi cc im cc i v im cc tiu lp thnh mt tam gic u.

Bi tp 17*. Cho hm s 4 2 42 2y x mx m m c th ( ). Vi

nhng gi tr no ca th th ( ) c ba im cc tr, ng thi ba

im cc tr lp thnh mt tam gic c din tch 4S .

92

Bi tp 18*. Cho hm s 4 22 1y x mx m c th ( ). Vi nhng

gi tr no ca m th th ( ) c ba im cc tr, ng thi ba im cc

tr lp thnh mt tam gic c bn knh ng trn ngoi tip bng 1 .

Bi tp 19*. Cho hm s 4 22 2y x mx ( ). Tm cc gi tr ca

( ) c 3 im cc tr to thnh mt tam gic c ng trn ngoi tip i

qua im

3 9;

5 5D .

Bi tp 20*. Cho hai s dng tha mn . Tm GTNN ca

biu thc 1 1

x yP

x y.

Bi tp 21*. Cho hai s dng thay i tha mn 5

4x y . Tm

GTNN ca biu thc 4 1

4S

x y.

Bi tp 22*. Cho , 0, 1x y x y . Tm GTLN, GTNN ca biu thc

2 2(4 3 )(4 3 ) 25S x y y x xy .

93

CHUYN II. TIM CN V S TNG GIAO CA HAI TH HM S

V D.

V d 8. Cho hm s

2( )

1

xf x

x. Gi l ng thng qua ( ) v c

h s gc . Tm ct ( ) ti hai im phn bit sao cho

thuc hai nhnh khc nhau ca ( ) v .

GII. PT ng thng ( 1)y k x . PT honh giao im ca ( ) v

2( 1)

1

xk x

x 2 (2 1) 2 0 ( 1).kx k x x (1)

t 1 1t x x t . Khi (1) tr thnh 2 3 0.kt t (2)

ct ( ) ti hai im phn bit thuc hai nhnh khc nhau

(1) c 2 nghim 1 2,x x tho 1 21x x

(2) c 2 nghim 1 2,t t tho 1 20t t 3 0 0k k . ( )

V A lun nm trong on MN v 2AM AN nn

2AM AN 1 22 3x x (3)

p dng nh l Viet cho (1) ta c:

1 2 1 22 1 2

(4), (5)k k

x x x xk k

.

T (3), (4)

1 22 1

;k k

x xk k

. Thay vo ( ) ta c 2

3k .

V d 9. Cho hm s 3 23 9y x x x m c th ( ). Tm tt c cc gi

tr ca tham s m th hm s cho ct trc honh ti 3 im phn

bit c honh lp thnh cp s cng.

94

GII. th hm s ct trc honh ti 3 im phn bit c honh lp

thnh cp s cng

Phng trnh 3 23 9 0x x x m c 3 nghim phn bit lp thnh

cp s cng

Phng trnh 3 23 9x x x m c 3 nghim phn bit lp thnh

cp s cng

ng thng y m i qua im un ca th ( )

11 11.m m

V d 10. Cho hm s 4 2 1y x mx m c th l mC . nh

th mC ct trc honh ti bn im phn bit.

GII. Phng trnh honh giao im ca ( ) vi trc honh

4 2 1 0.x mx m (1)

t 2 , 0t x t . Khi : (1) 2 1 0t mt m (2)

1,

1.

t

t m

YCBT (1) c 4 nghim phn bit (2) c 2 nghim dng phn

bit 0 1 1m

1,

2.

m

m

V d 11. Cho hm s 4 22( 1) 2 1y x m x m c th l ( ) nh

m th ( ) ct trc honh ti 4 im phn bit c honh lp

thnh cp s cng.

95

GII.

Xt phng trnh honh giao im 4 22( 1) 2 1 0.x m x m (1)

t 2 , 0t x t th (1) tr thnh 2( ) 2( 1) 2 1 0f t t m t m .

( ) ct ti 4 im phn bit th ( ) 0f t phi c 2 nghim

dng phn bit

2' 0 1

2 1 0 2

02 1 0

mm

S m

mP m

( )

Vi ( ), gi 1 20 t t l 2 nghim ca ( ) 0f t , khi honh giao

im ca ( ) vi ln lt l: 1 2 2 1 3 1 4 2; ; ; .x t x t x t x t

Ta c 1 2 3 4, , ,x x x x lp thnh CSC

2 1 3 2 4 3 2 19 .x x x x x x t t

1 9 1 5 4 1m m m m m m

4,5 4 4

45 4 4 .

9

mm m

m m m (tho (*))

Vy

44; .

9m

V d 12. Cho hm s Bin lun theo s nghim ca

phng trnh

2 2 21

mx x

x .

GII. t ( )

Ta c

2 22 2 2 2 1 , 1.1

mx x x x x m x

x

96

Do s nghim ca phng trnh bng s giao im ca th hm

s 2 2 2 1 ( ')y x x x C v ng thng , 1.y m x

Vi

2 ( ) 12 2 1( ) 1

f x khi xy x x x

f x khi x nn 'C bao gm:

+ Gi nguyn th ( ) bn phi ng thng 1.x

+ Ly i xng th ( ) bn tri ng thng 1x qua . Da vo th ta c:

v nghim 2 nghim

kp

4 nghim phn

bit

2 nghim phn

bit

V d 13. Cho hm s ( ) ( ) Da vo th ( )

hy bin lun theo m s nghim ca phng trnh

4 28cos 9cos 0x x m vi [ ]

GII.

Xt phng trnh: 4 28cos 9cos 0x x m vi [0; ]x (1)

t cos ,t x phng trnh (1) tr thnh: 4 28 9 0t t m (2)

V [0; ]x nn [ 1;1]t , gia v c s tng ng mt i mt,

do s nghim ca (1) v (2) bng nhau.

Ta c 4 2(2) 8 9 1 1t t m (3).

Gi ( ) [ ] v

Phng trnh (3) l phng trnh honh giao im ca ( ) v .

97

Ch rng ( ) ging nh th ( ) trong min

Da vo th ta c kt lun sau:

0m 0m 0 1m 81

132

m 81

32m

81

32m

v

nghim 1 nghim 2 nghim 4 nghim 2 nghim

v

nghim

V d 14. Cho hm s 3 2( ) 3 2y f x x x c th ( ).

a. Tm trn ng thng nhng im m t k c n ( )

ba tip tuyn.

b. Tm trn ng thng nhng im m t k c n ( )

hai tip tuyn vung gc vi nhau.

GII. Hm s xc nh trn v ( )

a. Gi ( ) l im nm trn ng thng c phng trnh

ng thng qua , h s gc c phng trnh

( )

ng thng ny v ( ) tip xc nhau khi v ch khi h sau c nghim:

{ ( )

T h trn, ta c phng trnh

( ) ( )

( ) ( )

T k c ba tip tuyn n ( ) khi v ch khi phng trnh ( ) c

ba nghim phn bit, hay phng trnh ( ) c hai nghim phn bit khc

0; tc l

98

{ ( )

b. S dng cu b, ta thy rng khi th Do , t k c

n ( ) hai tip tuyn vung gc vi nhau khi v ch khi phng trnh ( )

c hai nghim phn bit sao cho tch ca chng bng , tc l

{

( )

V d 15. Cho hm s

2 1( )

1

xf x

x( ) Gi l giao im hai ng tim

cn ca th ( ).

a. Gi 0 0 0( ; ) ( )M x y C . Tip tuyn ca ( ) ti ct hai ng TC ti

hai im phn bit v . Chng minh 0 l trung im . Tm v tr

ca 0 chu vi tam gic nh nht.

b. Tm cc im thuc ( ) sao cho tip tuyn vi ( ) ti ct cc

ng TC ca ( ) ti sao cho ng trn ngoi tip tam gic c

din tch nh nht.

c. Chng minh rng mi tip tuyn ca ( ) to vi hai ng TC mt

tam gic c din tch khng i.

d. Tm thuc ( ) sao cho tip tuyn ca ( ) ti vung gc vi

ng thng .

GII. Giao im ca hai tim cn l ( )

a. Ly

00 0

0

2 1;

1( )

xxM C

x. Tip tuyn ca ( ) ti c phng trnh

dng

99

( ) ( )

Tip tuyn ct tim cn ng v tim cn ngang ln lt ti

(

) ( )

R rng v nn l trung im

Trc ht, ta c

| | | |

Chu vi tam gic

Du bng xy ra khi v ch khi

Vy, chu vi tam gic nh nht khi

(

)

b. V tam gic vung ti v l trung im nn ng trn

ngoi tip tam gic c bn knh , do c din tch

[( )

( ) ]

Du bng xy ra khi v ch khi ( ) .

Lc ( ) hoc ( )

c. Tip tuyn ti im ct hai tim cn ca ( ) ti to thnh

tam gic c din tch

100

d. Gi ( ) ( ) suy ra

2 1

1

ab

a ( )

PTTT ca ( ) ti

21 2 1

( ) .( 1) 1

ay x a

a a

PT ng thng 21

( 1) 2.( 1)

y xa

Tip tuyn ti vung gc vi nn ta c

2 21 1

. 1( 1) ( 1)a a

0 1,

2 3.

a b

a b

Lc ( ) hoc ( )

V d 16. Cho hm s 3 2(1 2 ) (2 ) 2y x m x m x m (1). Tm tham

s m th ca hm s (1) c tip tuyn to vi ng thng

gc , bit 1

cos26

.

GII. Gi l h s gc ca tip tuyn, suy ra tip tuyn c VTPT

1 ( ; 1).n k

ng thng c VTPT 2 (1;1)n .

Ta c

1 2

21 2

. 1 1 3 2cos .

. 2 326 2 1

n n kk k

n n k

YCBT tho mn t nht mt trong hai phng trnh sau c nghim:

3

2

2

3

y

y

2

2

33 2(1 2 ) 2

2

23 2(1 2 ) 2

3

x m x m

x m x m

/1

/2

0

0

2

2

8 2 1 0

4 3 0

m m

m m

101

1 1;

4 2

3; 1

4

m m

m m

1

4m hoc

1.

2m

V d 17. Cho hm s

2(2 1)

1

m x my

x. Tm th ca hm s tip

xc vi ng thng .

GII. th tip xc vi ng thng y x th

2

2

2

(2 1) (

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K12_GT_CI_LT Ung Dung Dao Ham