BI TON TNG QUT

kho st hm s sau

3 2y f(x) ax bx cx d,(a 0)

Ta ln lt c

Tp xc nh: D

Gii hn 3 2x x

,a 0lim f(x) lim (ax bx cx d)

,a 0

o hm 2y ' 3ax 2bx c

2y ' 0 g(x) 3ax 2bx c 0

Ta c 2' g(x) b 3ac

Nu ' g(x) 0 hm s khng c cc tr

Nu ' g(x) 0 hm s c 2 cc tr

Lu : hm s bc 3 khng c trng hp c 1 cc tr

o hm cp 2 y'' 6ax 2b

by '' 0 6ax 2b 0 x

3a

Lp bng bin thin

Kt lun

Ty theo du ca h s a v du ca 2b 3ac m ta c kt lun khc nhau v cc tr v cc khong ng bin nghch bin ca hm s. th

hm s nhn 2 3

2

b 27a d 9abc 2bU ,

3a 27a lm im un v cng l

tm i xng ca th

th:

Ty thuc vo bng bin thin m ta c 6 dng th khc nhau ca

hm s bc 3.

Mt s tnh cht ca hm bc 3 cn nh l

Tnh cht 1: hm s ng bin (nghch bin) trn khi v ch khi

2

a ( )0

b 3ac 0

Chng minh: da vo vic bin lun du ca y ' v nh l o du tam

thc bc 2. Ta c ngay iu phi chng minh.

Tnh cht 2: hm s c cc i v cc tiu khi v ch khi 2b 3ac 0

v nu gi 0x l cc tr ca hm s th gi tr cc tr ca hm s l

2

0 0

6ac 2b 9ad bcy(x ) x

9a 9a

Chng minh: hm s c 2 cc tr th phng trnh y ' 0 c 2

nghim phn bit. Lp bit thc delta ta c ngay iu kin l y ' 0.

Gi 0x l cc tr ca hm s ta c

0y '(x ) 0 do ta c

23 2 2

0 0 0 0 0 0 0

2

0 0

3ax b 6ac 2by(x ) ax bx cx d (3ax 2bx c) x

9a 9a9ad bc 3ax b 6ac 2b 9ad bc

y '(x ) x9a 9a 9a 9a

20

6ac 2b 9ad bcx

9a 9a

Do ta c 2

0 0

6ac 2b 9ad bcy(x ) x

9a 9a

Tnh cht 3: th hm s bc 3 nhn im un U lm tm i xng

Tht vy, tnh tin h trc ta v im un U ta c ngay cng thc di trc l

2 3

2

bx X

3a27a d 9abc 2b

y Y27a

Thay vo hm s ta s c

3 22 3

2

23

27a d 9abc 2b b b bY a X b X c X d

3a 3a 3a27a3ac b

Y aX X3a

Nhn xt hm s 2

3 3ac bY aX X3a

l mt hm s l nn th

hm s s nhn im U lm tm i xng.

Tnh cht 4: tip tuyn vi th hm s ti im un c h s gc nh

nht khi a 0 v c h s gc ln nht khi a 0.

Ta c h s gc ca tip tuyn bt k vi th hm s ti im 0

x x

l

2 22 2

0 0 0 0 0

22 2

20

2

min 0

2

max 0

b b 3ac bk y '(x ) 3ax 2bx c 3a x 2 x

3a 9a 3a

3ac bk , a 0b 3ac b 3a3a x3ac b3a 3a

k , a 03a

3ac b ba 0,k x

3a 3a3ac b b

a 0,k x3a 3a

Do ta c iu cn phi chng minh

Tnh cht 5: th hm s ct trc honh ti 3 im cch u nhau hay

honh cc giao im lp thnh cp s cng khi v ch khi 2 327a d 9abc 2b 0 .

Tht vy ta c phng trnh honh giao im gia th hm s v

Ox l

3 2ax bx cx d 0

Gi s th hm s ct Ox ti 3 im phn bit th phng trnh trn

s c 3 nghim tha mn 1 2 3

1 3 2

x x x

x x 2x. Mc khc theo nh l Viete

ta c 1 2 3 2 2

b b bx x x 3x x

a a 3a

Mc khc 2x l nghim ca phng trnh 3 2ax bx cx d 0 do

ta c

3 2

3 2

2 2 2

2 3

U

b b bax bx cx d 0 a b c d 0

3a 3a 3a

27a d 9abc 2b 0 y 0

Do ta c iu phi chng minh.

Lu : iu kin trn ch l iu kin cn khng phi l iu kin nn

nu bi ton yu cu tm iu kin tham s th hm s tha mn

tnh cht trn th sau khi tm c tham s cn phi kim tra li

Tnh cht 6: nu th hm s ct Ox ti 3 im c honh lp thnh

cp s nhn th 3 3ac b d 0

Tht vy ta c phng trnh honh giao im ca th hm s

cho vi Ox l

3 2ax bx cx d 0

V th ct Ox ti 3 im c honh lp thnh cp s nhn nn ta c

phng trnh 3 2ax bx cx d 0 s c 3 nghim tha mn 2

1 3 2x x x . Mc khc theo nh l Viete ta c

2

1 2 2 3 3 1 1 2 2 3 2

2 1 2 3 2 2

c cx x x x x x x x x x x

a ac b c c

x (x x x ) x xa a a b

V 2x l nghim ca phng trnh 3 2ax bx cx d 0 nn ta c

3 2

3 2

2 2 2

c c cax bx cx d 0 a b b d 0

b b b

3 33 3

3

ac b d0 ac b d 0

b

Do ta c iu phi chng minh. Lu cng nh tnh cht 5, tnh cht

6 cng ch cho ta iu kin cn.

Tnh cht 7: gi th hm s ct Ox ti 3 im phn bit, gi 1 2S ,S l

din tch hnh phng gii hn gia th v Ox (phn nm pha trn v

pha di trc Ox ) Nu 1 2S S th im un U thuc trc honh

Ta c nu im un thuc trc honh th 1 2S S , bng phng php

dch chuyn th ta s c iu cn chng minh

Tnh cht 8: gi s hm s 3 2y ax bx cx d c th l (C).

Trn (C) ta ly 3 im M,N,P sao cho M,N,P thng hng. Tip tuyn

vi (C) ti M,N,P ln lt l M N P(t ),(t ),(t ). Cc tip tuyn

M N P(t ),(t ),(t ) ct th (C) ti 3 im khc l I,J,K th ta s c 3 im

I,J,K cng thng hng

Ta c ta cc im M,N,P ln lt l 3 2(m,am bm cm d), 3 2(n,an bn cn d) v 3 2(p,ap bp cp d) khi ta c 3 im

M,N,P s ng vi 3 s phc sau 3 2Mz m i(am bm cm d) ,

3 2

Nz n i(an bn cn d) v 3 2

Pz p i(ap bp cp d).

V cc im M,N,P thng hng nn ta c

M N M N

P N P N3 3 3 3 3 3

2 3 3 2 2 2

z z z zIm 0

z z z z

(am an bm bn cm cn )(p n)

(p n) ( an ap bn bp cn cp)

3 3 2 2

2 3 3 2 2 2

3 3 3 3 3 3

3 3 2 2

(m n)( an ap bn bp cn cp)

(p n) ( an ap bn bp cn cp)(am an bm bn cm cn )(p n)

(m n)( an ap bn bp cn cp) 0

(n p)(m p)(m n)(am an ap b) 0

V cc im M,N,P phn bit nn m n p do ta c

(m n)(n p)(m p) 0 am an ap b 0

Phng trnh tip tuyn vi (C) ti M l 2 3 2

M(t ) : y (3am 2bm c)(x m) am bm cm d

Phng trnh honh giao im ca M(t ) v (C) l

3 2 2 3 2

2

I

2 3 2 2

I

2 3 2 2

ax bx cx d (3am 2bm c)(x m) am bm cm d

(x m) (2am ax b) 0

x m2am b

x2am bax

a8a m 8abm 2acm 2b m ad bc

ya

2am b 8a m 8abm 2acm 2b m ad bcI ,

a a

Tng t ta cng s c

2 3 2 22an b 8a n 8abn 2an 2b n ad bcJ ,

a a

2 3 2 22ap b 8a p 8abp 2ap 2b p ad bcK ,

a a

Do ta s c 3 s phc tng ng vi 3 im I,J,K l

2 3 2 2

I

2 3 2 2

J

2 3 2 2

K

2am b 8a m 8abm 2am 2b m ad bcz i

a a2an b 8a n 8abn 2an 2b n ad bc

z ia a

2ap b 8a p 8abp 2ap 2b p ad bcz i

a a

Ta c

2I J

K J

z z 4(m p)(m n)(am an ap b)aIm 0z z (m,n,p)

Do ta c I,J,K thng hng.

Tnh cht 9: bin lun s nghim ca phng trnh bc 3

Xt phng trnh bc 3 sau 3 2ax bx cx d 0

Ta c 2 cch lm thun i s-gii tch nh sau

Cch 1: cch lm thun i s

Gi s ta bit c 0

x x l mt nghim ca phng trnh trn khi

ta c

2 2

0 0 0 0

02 2

0 0 0

(x x )[ax (ax b)x ax bx c] 0

x x

g(x) ax (ax b)x ax bx c 0

Do ta c

3 2ax bx cx d 0 c 1 nghim duy nht th iu kin l

phng trnh g(x) 0 v nghim hoc c nghim kp l 0x . Do ta

c iu kin s l

2 2 2

0 0

2 2 2

0 02

0 0 0

3a x 2(bx 2c)a b 0g(x) 0

g(x) 0 3a x 2(bx 2c)a b 0

g(x ) 0 3ax 2bx c 0

Vy iu kin l

2 2 2

0 0

2 2 2

0 02

0 0

3a x 2(bx 2c)a b 0

3a x 2(bx 2c)a b 0

3ax 2bx c 0

phng trnh 3 2ax bx cx d 0 c 2 nghim th phng trnh

g(x) 0 phi c 2 nghim phn bit v mt nghim trng vi 0x . Do

ta c ta c iu kin l

2 2 2

0 02

0 0 0

g(x) 0 3a x 2(bx 2c)a b 0

g(x ) 0 3ax 2bx c 0

Vy iu kin l

2 2 2

0 02

0 0

3a x 2(bx 2c)a b 0

3ax 2bx c 0

phng trnh 3 2ax bx cx d 0 c 3 nghim th phng trnh g(x) 0 phi c 2 nghim phn bit v khng c nghim no trng vi

0x . Do ta c iu kin l

vy iu kin cn tm l

2 2 2

0 02

0 0

3a x 2(bx 2c)a b 0

3ax 2bx c 0

Cch 2:cch lm thun gii tch

Xt hm s 3 2y ax bx cx d

Tp xc nh D

o hm 2y ' 3ax 2bx c

2y ' 0 g(x) 3ax 2bx c 0

phng trnh 3 2ax bx cx d 0 c 1 nghim th hm s 3 2y ax bx cx d khng c cc tr hoc c 2 cc tr cng nm

v mt pha ca Ox do ta c iu kin l

2

2

2 2 3 3 2 2cd ct

b 3ac 0' g(x) 0

' g(x) 0 b 3ac 0

y y 0 27a d 18abcd 4ac 4b d b c 0

Vy iu kin l

2

2

2 2 3 3 2 2

b 3ac 0

b 3ac 0

27a d 18abcd 4ac 4b d b c 0

phng trnh 3 2ax bx cx d 0 c 2 nghim th hm s 3 2y ax bx cx d c 2 cc tr v mt trong 2 gi tr cc tr phi

bng 0 do ta c iu kin l

2 2 2

0 02

0 0 0

g(x) 0 3a x 2(bx 2c)a b 0

g(x ) 0 3ax 2bx c 0

22 2 3 3 2 2cd ct

' g(x) 0 b 3ac 0

y y 0 27a d 18abcd 4ac 4b d b c 0

Vy iu kin l

2

2 2 3 3 2 2

b 3ac 0

27a d 18abcd 4ac 4b d b c 0

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

hm s 3 2y ax bx cx d phi c 2 cc tr v 2 gi tr cc tr

tng ng tri du vi nhau do ta c iu kin l

2

2 2 3 3 2 2cd ct

' g(x) 0 b 3ac 0

y y 0 27a d 18abcd 4ac 4b d b c 0

Vy iu kin cn tm l

2

2 2 3 3 2 2

b 3ac 0

27a d 18abcd 4ac 4b d b c 0

Da vo tnh cht ny ta c th m rng ra

iu kin phng trnh bc 3 c 3 nghim dng l

2

2 2 3 3 2 2

b 3ac 0

27a d 18abcd 4ac 4b d b c 0

ad 0,ac 0,ab 0

iu kin phng trnh bc 3 c 3 nghim m l

2

2 2 3 3 2 2

b 3ac 0

27a d 18abcd 4ac 4b d b c 0

ad 0,ac 0,ab 0

iu kin phng trnh bc 3 c 3 nghim trong c ng 2 nghim

dng v nghim cn li m l

2

2 2 3 3 2 2

b 3ac 0

27a d 18abcd 4ac 4b d b c 0

ad>0,ab0,ab>0

Trn Quang Minh K2008-2011 Thpt L T Trng Nha Trang Khnh Ha