Nh^ gi^o uu
tij
-
TTi^c
siToAn hpc
NGUYEN VAN THONG
(Chu
bifen)
Gl^ng
vi«n
chfnh
-
Thac
sITo^n hpc
NGUYEN VAN MINH
Ph6 gi^o su
- mn
si:
NGUYEN VAN HIEU
••a
LIIYeNGIIUIliniUIICKVTHIIilUHOCSMliN
TOAN
HOC
ii@®
Ddnh
cho hpc
sinh
12
luy^n
thi
BH-CB

^
Biin
so$n
theo
n0i
dung djnh hudng
^
i
mdi
cua
BO
Gi^o
Due
&
B^o Tao
MZ
THLT
MIEN
TiWH BINH THUAN
DVL
J
My^6' /I5 I
NHAXUJTBiiNTONGHQPTHiiNHPHOHOCHlMINH
Left
noi ddu
Chiing
toi
la cac
anh em ruot
thjl,

muo'n vie't quyen
sach
nay cho the
he
sau on
tap
de
chuan
bi
thi v^o dai hoc, vi bU'dc v^o
triTdng
dai
hoc,
ngtfcfi
hoc
sinh b^l
dau mot ddi
song
mdi
vk c6
tiTdng
lai
tiTdi
sang
khi chpn
dtfdc
mot
triTcJng
dai
hoc

tot,
ch^c
ch^n r^ng
triT^ng
nay se
chon diem
cao.
NhiT
vay trong
qua
trinh
on
luyen
cac
em can mot tai lieu tUcfng thich. Dieu nay
se
diTdc
thoa man neu cAc em
chiu
kho on tap
theo
cac
chuyen
de mh
chung toi bien
soan,
ch^c
ch^n thi
se
dai

diTpc diem cao.
Quyen sdch nay gom
20
chu
de
trong diem
theo
cau true
de
thi cua
Bo
giao
due hang nam,
cac
vi du duTa
ra
ttfcfng doi kh6
va c6
hiTdtng dan giai. Tie'p
theo
la
28
bp
de
thi cho
cac
khoi A,
B, D
cua tri/cfng chuyen
Le

Quy Don
- Da
N^ng
dilng
de
thi thijr tiTng nSm qua
66
danh gid diTdc hpc sinh, nit ra
kinh
nghiem
giang day. TriTdng
Le
Quy Don
- Da
NSng hang nam deu
c6 ti 16
dau vao
cac
trirdng
dai hpc
la 100%.
Rieng nam
2010 c6 so
hpc sinh dat diem
cao
diTcfc xep
thu"
4
trong bang xep
hang cua

Bp
giao due va Dao tao
Cic chuyen
de
chiing toi deu
soan
tilf
can
ban den nang
cao,
dp kho dii
de
cho
cac
hpc sinh khd
va
gioi
tU'
luyen, neu thong hieu taft
ca,
ch^c ch^n ring cac
em
se
giai du'pc
de
thi Dai hpc mpt cdch
de
dang,
nh\i
chiing toi

da
tilfng
luyen
rat nhieu
em do
thu khoa
cic
tru'dng Dai hpc danh tieng. Dieu quan trong hdn
nifa
vdi
each
vie't ciia mOt gido vien day chuyen Todn lau
nim, nen
each
gpi
md,
din d^t mang dam
n6t
t\i
duy nang
cao, hpc
sinh
se
diTcJc
phat trien
kha
nSng
Toan hpc khi hpc quyen sdch n^y.
Mong
muon ciia chiing toi l^m

the nko de c^c em
tif hpc to't mon Toan,
va
kha nang thi dau v^o
cdc
tri/dng
Dai
hpc
Idn phai nh5
vko
sir kien tri
cua cic
em
"c6
cong mai s^t
c6
ng^y nen
kim"
cd nhan day that
diing
vay.
Quyen
sich
nay
cung
1^ ky
nipm
3 nim cic em hpc
Toan vdi thay: Bich
Lien,

PhiTdng Thao, Anh ThiT, Mai HiTdng, Thiiy Hien, Th^o Uyen.
Cic
em
da
giiip
thay chinh
sufa
ban thao, mpt
cich
nhipt
tinh
trdch nhipm.
Chiic
c^c
em
se
th^nh cong my man trong ky thi s^p
idi
Chu bien: Nguyen VSn Thong
(To
trirdng
To Todn
trirdng
chuyen
Le
Quy Don
-
Da
NJng)
Nhd

sdch
Khang
Vi$t
xin
tran
trgng
giai
thi^ tai Quy dgc gia va xin
Idng
nghe
tnoi
y
kien
dong
gop, de
cuan
sdch
ngdy
cdng
hay han, bo ich
hart.
Thu xin giH
ve:
.„,.^
_
Cty
TNHH
Mpt Thanh Vien
-
Djch Van Hoa Khang Vi?t.

71,
Dinh
Tien Hoang,
P.
Dakao.
Qu^n
1, TP. HCM
Tel: (08)
39115694
-
39111969
-
39111968
-
39105797
-
Fax: (08)
39110880
Hoac
Email:
I
mill inx-r
z^r TWM
I\FM
'K^mdt
m
MJOl
C«0
TRONG
DI^M

GI(3l
HAN
vA
TfNH
UfeN
TUC C6A MOT HAM
S6
I.
T6M TAT LI THUYET
1.
Cac
djnh
If
ve
gidi
han
••1
('
Gia sur lim f(x)
=
L,
va
lim g(x)
=
Lj, khi do
X-^X()
X-+X,)
lim [f(x)
+
g(x)]=

lim
f(x)+ lim g(x)
=
L|+L2
x->X()
"->"() "-^^o
lim
[f(x)-g(x)]=
lim
f(x)-
lim
g(x)
=
L,-L2
x->X()
"-+"() x-^xo
lim
[f(x).g(x)]=
lim
f(x).
lim
g(x)
=
L|.L2
x^X() x-^xo
''->X()
=' 'v,V'
lim
f(x)
,

lim
X-^X()
f(x)
g(x)
X^X()
_
^1
lim
g(x)
L2
vdi
X-*X()
Neu
3s > 0
sao cho f(x) < g(x)
Vx
€ (xo
- e;
x,)
+ s)
va ton tai
1 , ,
lim
f(x), lim g(x) thi
lim
f(x)
<
lim
g(x)
x->X()

''-»X()
x^XQ
x->X{)
li
•
a.
NguyinIfgidinank?p:
Neu
3E
> 0:
f(x)
<
h(x) < g(x)
Vx e
(x„
-
E; X„
+ e)
\}
va lim
f(x)=
lim g(x)
= L
thi lim h(x)
=
L.
x^X() x-»X() x-»X()
b.
Cdc
dang

gidi
han
dQc
bi$t
lim
=
1; lim (1
+ x)x
=
e
X->X()
X X^XO
I,,,,,.,.,.;,.,:
lim
X->X()
=
e; lim
e^-l
2.
Gidi
h^n dang vd
djnh:
—
=
1; lim
x-»X()
X
x->X()
0
ln(l

+ X)
=
1
P(x):dathu'c, P(x„)
= 0 >
P(x)
a.
Dang
1.
I =
lim
—^
vdi
<.
x^x„
Q(x) [Q(x): da thiTc, Q(Xo)
= 0
PhiTdng ph^p:
7 —- .
i'
1=
Hm
lim
i^i^^iolW
^
P^^P^ vdiQ,(x«)^0
x^xoQ(x) x->Xo(x-Xo)Qi(x) X^X()Q,(X)
Qi(Xo)
. J I
IWtg,

„ |,W
Ne'u Pi(x„) = Qi(x,)) = 0 thi phan
tich
tie'p •
0
P,(x) =
(x-Xo)P2(x)
Q,(x) =
(x-Xo)Q2(x)
Qu^
trinh
khuT
dang v6
dinh
^ la qua
trinh
khur
cac nhan tuT chung (x-Xo)'' se
difng
lai khi nhan du'dc
gidi
han xac
dinh
ttfc
la Qk
5^
0.
Khid6I=
lin ^=limAW^M^
>'^''()Q(x) x^x„Q,,(x) Qk(xo)

'f(X(,)
= g(x„) = 0 OO.
f(x),g(x)
chtfa can thtfc dong bac
b.
Dang
2.
lim
vdi
\
x->xo
g(x)
PhiTdng phap: SuT dung cdc hKng d^ng thtfc de nhan lien hdp 6
tuT
va mau
nh^m
true cac nhan
tijr
(x - x,,) ra khoi can thtfc.
A-B
A
+ B
A-B
2"M/A+2n+^^
A
+ B
c. 5.1 = hm vdi
' x^xog(x) [f(x) chtfa can khong dong bac
^^"0
g(x) x^XQ

g(x)
Bi6n
doi
1= lim -
x->xo
'?yu(v)-ci-rjyv(x)-c
g(x)
- lim
X-+XO
.g(Xo)
= 0
•^uW-c 5!/v(x)-c
g(x)
g(x)
Den day cac
gidi
han diTdc
tinh
theo dang 2.
3. Gi6i hain v6 djnh —
00
PhiTdng ph^p: X6t I = lim ^ vdi P(x), Q(x) la cac da thi?c
hoSc
cic ham
x-»xo
g(x)
daj s6. Gpi bac P(x) = p; bac Q(x) = q v^ m =
min(p,
q), khi d6 chia ca va
mau cho x" ta c6 ket luan sau:

C6ngiyTNi:ii
:
;.W//A/.U;.^
,UI
• Ng'u p < q thi ton tai
gidi
han
• Neu p > q thi khong ton tai
gidi
han
4. Gi6i han dang v6 dinh « -
oo
Phifdng
phap: Bien ddi diTa ve dang
gidi
han — '
1 0>t
Tim
gidi
han sau
Vx +Vx -Vx
lim
x-++<»L
(x
+Vx j-x
=
lim . = lim
J
x^+«>^x + ^/^+^/x "
5. Gidi han dang v6 djnh

oo
. 0 , .
Phifdng
phap: Du'a ve dang v6
djnh
—
00 ~. ^'J
Ching han, tim
gidi
han
Vx^ + l-x
lim
X->+oo
.
=
lim
X-»+a>
Vx^ +1 +:
lim
mil
tioub fi?. im
x
l
+
-^+l
X
1
2
6. Gidi h^n dang vd djnh ham lUging giac
PhtTdng phap: Suf dung cac ket qua

gidi
han cd ban sau dSy:
,. sinx , ,. X ,
• hm = 1; hm =
1
x->()
sinx
• lim
x->0
X
sin
ax
V
=
lim
x->0
X x-»0
sin
ax
I
ax
sin
ax
,. sinax ,. sinax
=
a. hm a
=>
hm = a
x^O
X „ r

x->()
ax
hm
sinax
=
Um
ax
-^ =
^^lim
mil
—
iiiii
.—:—-— — — —• 11111 — —
x->()sinbx
x-»()bx sinbx b x-+()sinbx b
smax
a , .
=
-(a,
beR*)
bx
lim
tan
ax
a sinax ,. tanax
hm
. = a => hm = a
x-»o
x
x^ocosax

ax
tanax
• Hm
tanax
,. ax ax a ,.
=
hm —• = - hm
x->0
X
tanax a
x->otanbx
x->{)bx
tanbx b x->()tanbx b
bx
Sinax
hm
sinax , sinax ,. ax .ax ^
hmcosbx.
=
hm—.cosbx.
. .
x-*()
tanbx x->() sinbx x->obx sin Dx b
bx
Luyfn
gidi
di
truOc
kp ihi DH 3
miin

Bdc,
Trung,
Nam
Todn
hoc -
Nguyin
Van
ThOng
7.
Gidi
han
dang
v6 djnh 1°
Phifdng phap:
a. Sijfdung:
lim(l
+ x)" =e; lim
x->()
x^+«>
b. Xct lim uCx)"*"'
CO
dang
T
X^X()
• •
f
1
1
+ -
t'l

I ,,
" US;'
fili'd
i&ili
ffH'T
=
e ,,,,
Bie'n
doi lim
X-»X()
•
''1
l
+
(u-l)"-'
lim
(u-ir
X
\ •+ X
8.
Gidi
han d?ng v6 djnh cua ham mu va Idgarit: ^
^^'^ *
Phifdng
phap:
S^rdung
lim^ ^ = 1; lim ^ =
1
x-»()
X

II.
BAI TAP
MINH
HQA
x->()
X
Bai
1. Tim
gidi
han
lim f(x),
vdi r(x) =
X->1
x^-1
Ta
CO
lim
x^l
Htfdng din
sial
Vs-x^ -2 \/x^ + 7-2^
x^-l x^-l
(1)
Matkhac:
hm
=
lim -
X^l
X -1
"-l/v^

1-x-^
Va
lim
Vx"
+ 7-2
=
lim
-fx^+x
+ l)
=
lim \ ' (2)
X->1
x^-l
x->l
X -1 "->!
^"'(x^-lWx^+vf+2^/?77
=
lim
x^l
x^+7+4
1
3
v2
x^+7
x^
+7 + 4
12
(3)
•a
Thay (2), (3) vao (1). la difdc L = — = -—

8 12 24
.
. ™ ,. x'^ -4x^ +4x -3
Bai2,
Tim
lim—
"-^^
x 3x
Hifdng din giai
Ta
CO
lim '^^^^ = ^JZ^ = 1 >
x^3
(X-3)X
x-»3 X
.
vl^-1
Bai
3. Tim hm
x->0
X^
I-
Hifdng d§n giai - - -
Ta c6: lim
(l
+
x2)-l
x (.^2
-
= lim

—
= iim , , =
—
3/^1777+^/i77^^1
''-*"^(i+x2)+^/iT^+i
3
Bai
4. Tim hm
x-»()
X
Hi/dng din giai
Ta
CO
,.
(2N/rT7-2)
+
(2-^/8^)
,.
hm
lim
x-»()
X x^O
X
=
Hm
x-^o
=
hm
x->()
2x

>/l + x + l) X[4 + 2W^ +
N/(8-X)^
,„;.,.
fijsi
^/^^
+ 1^4 + 2^ +
^(8-x)2J
""12 12
Bai
5. Tim lim
X-»+oo
.
(x + 1)'"" + (x +
2)'"*'
+
•••
+ (x + 9)'"" +
(X
+100)
l(X)
.100
Ta c6 lim -
X-»+QO
Bai
6. Tim
gidi
han lim
x-»l
x'""
+ 10x'" + 100'"

Hi/dng din giai
2
^
,^100 ^ ^^100 ^ j^^N
1
+
+ +
100
1
+
.100
19^ loo^ll^
+
^10+
^HX.
=
100
m
lim
X->1
m
Al-x'" l-xj
1-x"" 1-x"
Hi^dng dan giai
f_n. i_^] :;
,
(m, n e
N*,
m n) , ,
Luy(n

gUU
d6
trade
thi
DH
3
miin
Bdc.
Trung.
Nam
Todn
hoc -
Nguyen
Van
ThOng
lim
X-+1
lim
x-»l
lim
t
m-(l
+ x + x^+ +
x'""')
n-(l + x + x^+ + x""')
l-x""
1-x"
(l-x) + (l-x^) + + (l-x"'"') (l-x) + (l-x^) + + (l-x"-')
(1
- X)(l + X + x^ + +

x"'"')
(1 - x)(l + x + x^ + + x"~')
l
+ (l + x) + + (l + x + +
x"''^)
l + (l + x) + + (l + x + + x""^)
l
+ x + x2+ +
x"'-'
l
+ 2 + + (m-l) l + 2 +

+ (n-l)
1
+ x + x^ + + x" '
m(m-l) n(n-l) = '
m
m
m-1
n-1 m-n
2
2 2
I , •
„ %/cosax-Vcosbx ^, ,^ .
Bai
7.
Tinh
L = lim -^^ (a, b, c la cac so
thifc
khac khong) m, i

"-»(' sin cx
<•
m cac so tir
nhien
khac
nhau
Idn hdn 1. '' ^
Hifdng din giai
Bdde: lim
1-cos ax a
(1)
(l-jycosbx)-(l-".ycosax)
lim
r
x^O (cx)"
Ap
dung
(1):
smcx
= Hm
x-*()
= -^lim
C x-X)
1-cosbx
l-iVcosbx l-'^cosax
(cx)2 (cx)2
r
1-cos ax
l
+ !ycosbx+ + N/cos"-'bx

1
+
"Q^
cosax + + \/cos"' 'ax
bM aM
2 n 2 m
2c^
n
m ,
.1-
BaiS.Tim
lim
n->-H»
a
a a
cos—.cos— cos-
2"
;a^0
lAvtdng
dSn giai
A^aa
a a a aa
A = cos
—.cos — cos
= COS .cos r COS-T-COS —
2 2
1
^
2
sin

. a a
/sm
— cos
2^^
2^^
,n-l
2"
2"
2" j
a
a a
.cos r COS-T-COS—
,n-l
2 '
2
sin
2sm
rcos
2n-l
2"'
a
a
cos-^.cos—
2^
2
2"
2"
sin ^
2sin—cos—
2 2

sma
2"
2".sin-^
2"
a
i'.'Jir]
I',- jj •••
1- sma 2" sma ,,. .,
sma
2 sm— sm-
2"
2"
III.
BAI TAP Ti; LUYEN CO DAP SO. jj,, ,
cos
Bai
1. Tim
gidi
han sau: C = Hm —;
—
cosx
2
x-»o
sin(tanx)
Hifdng din giai
n
cos —cosx
u
C=
lim^iML

x->o
sm(tanx)
tanx
•
= = 0
Bai
2. Tim
gidi
han sau: I = Hm
. e cos x-1
x^O x"
HiMngdSngiai '"'^
I
= Hm
x->()
= 3-1=2.
e-^" -1 Icos^x + cos^x-l
•= Hm
x->()
3cos^
X.
3x2
ijji'j
i •
'
Sin
X ^
I
X ;
B^i

3.
Tinh
gidi
han I = lim
x-»()
1= lim
x-+()
e" -1
. e" -Vcosx
+ln(l
+ x^) ' •!
x^
2(l + V^)Yx2'l
HrfdngdSngiai
^^"'^ ln(l + x^)
+ -x
, 2 ,
4 ,
Luyfn
giil
di
trudc
thi
DH
3
miin
Bdc,
Trung.
Nam
Todn

hpc
-
NgiQ^n
Van
Tlidng
CAC
BAI
TOAN
KHAO
SAT HAM s6
I.
T6M TAT LI THUYET:
1.
Cac bu6c khao sat si/ bien thien va ve do thi cua ham so
Bifdc
1. Tim tap xac djnh cua ham so
Btfdc
2. Xet
sU'
bien thien cua ham so
-
Tim
gidi
han tai v6 cifc va
gidi
han v6 cifc (neu c6)
ciia
ham so. Tim cic
dircJng tiem can
ciia

do thi (neu c6). l> "
-
Lap
bang
bien thien cua ham so, bao gom:
Tinh
dao h^m cua ham so, xet
dau dao ham, xet chieu bie'n thien va tim ciTc tri cua ham so (neu c6), dien
.
cac ket qua vao bang. ^.^ ^
BMcJ.
Ve d6
thi
cua ham so. ^ ^ j
-
Ve cac du'dng tiem can cua do thi (neu c6).
-
Xac dinh mot so diem dac biet cua do thi,
chang
han tim diem uon, giao
diem
cua do thi
vdti
cac true toa dp (trong triTdng hdp do thi khong c^t cac
true toa dp
hoSc
vice tim tpa dp giao diem phtJfc tap thi bo qua phan nay),
-
Nhan xet ve do thj: Chi ra true va tam doi xuTng cua do thi (neu c6, khong
yeu cau chufng minh).

2. Diem uon cua do thj: la diem
U(X();
yo)
ciia
do thj sao cho lie'p tuye'n tai dd
di
xuyen qua do thi, tiJc la ton tai mot khoang (a; b) chuTa diem
Xo
sao cho
tren mot trong hai khoang (a; x,,)
hoSc
(X(i;
b) tiep tuye'n cua do thi tai diem
U
nam phia tren do thi, con tren khoang kia tiep tuyen n^m phia du'di do thi.
Ta thU'dng
suT
dung ket qua sau day de tim diem uon cua do thi:
Neu ham so y = r(x) cd dao ham cap hai tren mot khoang chuTa diem
Xo,
f"(xo)
= 0 va r'(x) doi dau khi x di qua diem Xo thi
U(X();
f(X()))
la mot diem
uon cua do thi ham so'y = f(x).
3. Giao diem cua hai do thi.
a. Phu"dng
trinh
(xac djnh) hoanh dp giao diem

ciia
hai do thj y = l(x) va
y
= g(x)
(ciing
ve tren mot mat phing tpa dp) la f(x) = g(x) (1)
b.
So nghiem cua (1) cung chinh la so giao diem cua hai do thi y = f(x) va
y
= g(x). Dac biet: PhiTdng
trinh
(1) cd nghiem (v6 nghiem) khi va chi khi hai
do thj
citt
nhau (khong cat nhau). i
c. Ta thudng gap u-irdng hpp phiTdng
trinh
(1) cd
dang
f(x) = m, trong dd m la
tham so va ham so f(x) khong chtfa tham so m. TriTdng hdp nay ta cd bai
C6ng
ty
TNHH
MTVDWH
Khang
VH
toan xdl giao diem cua do thi y = f(x) vdi drfdng thing y = m (vuong gdc vd
true tung va c^t true tung tai diem cd tung dp m).
4.

SI/
tiep
xuc cua hai dudng
cong.
•
a. Dinh
nghla:
Hai ham so' f(x) va g(x) tiep
xiic
nhau tai diem M(xo; y,,) neu M
la
mot diem chung cua chung va hai di/dng cong cd chung tie'p tuye'n vdi
nhau tai M.
b. Dinh
It:
Hai diTdng cong y = f(x), y = g(x) tiep xuc vdi nhau khi va chi khi he
ff(x)
= g(x)
phiTdng
trinh
an x sau day cd nghiem: \
[f'(x)
= g'(x)
(He phiTdng
trinh
nay la h? phiTdng
trinh
\ic dinh hoanh dp tiep diem cua
haidirdng).
' - •

II.
CAC BAI
TOAN
MINH
HQA.
Bai
1. Cho ham so y = f(x) = -x' + 2x^ + |x "^^
a. Khao sat h^m so.
b.
Tiep tuyen cua do thi (C) cua ham so tai goc tpa dp, cat (C) tai diem M.
Tinh
tpa dp diem M.
c. Bien luan
theo
k so giao diem cua (C) va du'dng thing (d): y = kx.
n
; I a Hi^ngdlngiai
a. Khao sat ham so vi£a>t,
l.Tapxacdinh:D = R -
1
2.
Sir bien thien:
a.
Gidi
han:
lim
y= lim
X-»+00
X->-H»
J/

^
o 2 5
-X
+2x +-X
-
-00
3
. 2 5 .
lim
y= lim (-x +2x +-x) = +oo
X->-QC
X->-<»
3
b.
Bang bien thien
y'
= -3x' + 4x + —, y' = 0 o
X
= —
hoac
x
3 ' 3 •
5
3 • ^ '
X
-00
-1/3 5/3
+00
y'
—

0
+
0
y
^
-8/27
100/27 ^
-00
Ham so dong bien tren khoang
f
\_5
to
Ob i;-!
Luy?n
gidi dS IruOc kp thi DH 3 miin Bdc, Trung, Nam Todn hgc - NguySn Vdn Th6n^
Ham so" nghich bien tren cac khoang
—;+oo
, . . 5 . 100
Ham so dat cifc dai tai x = - va y^^ = —
1
mms^datcirctieutai
x y^=—i.''>
a»arb
myii. oui lil
3.
D6^thi
(Hoc sinh W ve) = ^ _^^y. ^ g,^,^
^0^3^,}^
jj^H
^^J^^Q J|

+ Diem uo'n , -/.v i« ,r/-if>'t) ,,
^ . n 2 /2 46
y"
=-6x + 4, y" =0ox= Dieml -;—
3 , .3 2V J
hi
+ x =
0=>y=
—
^''^>''<
l-^^''''"
la
diem uon.
+ NhSn x6t: D6 thi nhan diem I
'2
46^
3'27
lam
tam doi xtfng.
b.
PhiTdng
trinh
tiep tuyen (A) cua (C) tai O (0; 0) la:
mi
y
=
f(0)(x-0)
+ 0= ^x
SB"
0*

PhiTdng
trinh
hoanh do
giao
diem ciia (A) v^ (C) IS:
"x
= 0
x
= 2
-x^+2x^+-x x o
-x^(x-2)
= 0c^
3 3
• x = 0=>y = 0
• x = 2=>y = —
3
r,
10^
'3
c. Phu-dng
tnnh
hoanh do
giao
diem cua (C) va (d) Ik:
-x''+2x^+|x =kx(l)ox
x^-2x + k
3J
= 0o
x
= 0

x2-2x + k 0
3
So'
giao
diem cua (C) va (d) li so nghiem cua phiTdng
trinh
(1). ^
Datg(x)=
x^-2x + k la tam thiJc bac hai
CO
A' = k
3 • 3
o
Tmyn^ /. A' < 0 o k > -
Khi
do (1)
CO
dung 1 nghiem <=> (C) va (d) c6 dung mot diem chung.
Tmyrtfi hap 2. A' = 0 <:> k = -
Khi
do (1)
CO
dung 2 nghiem <^ (C) vk (d) c6 diing 2 diem chung.
H
g
TrUdng
h<tp
3.
A'>0<:>k<
-

+
Ne'ug(0)
=
0<=>k=
- thi(l)c6dung2nghiem.
+ Neu g(0)
9t
0 o k ;t - thi (1) c6 dung 3 nghiem o (C) va (d) c6 diing 3
3 , ,1)/, . .,
diem chung.
Bai
2. Cho ham so y = f(x) = -x^ + 3x^ + 3(m^ - l)x - 3m^ -1 (1), m la tham so.
a. Khao sat sifbien thien va ve do thi ham
s6'(l)
khi m = 1.
b.
Tim m de ham so' (1) c6 cifc dai, ciTc tieu va cac diem cUc tri cua do thi ham
so (1) each deu goc toa do O.
HrfdngdSngiai .,4,)(„f, .
a. Khao sat ham so (Hoc sinh tif giai). „ , . >
b.
Ta c6 y' = -3x^ + 6x + 3(m^ -1) ; '
rx = l + m lv*i K;vM.!: • .Won
y'=0«
[x =
1
- m
Dieu
kien de ham so c6 ctfc dai, cifc tieu la y' = 0 c6 2 nghipm phan biet
om?<i0.

Goi A(l + m; -2 + 2m'); B(l - m; -2 - 2m') la hai diem ciTc
tri.
Ta c6 O each deu A va B o OA = OB
o (1 + m)^ + (- 2 + 2mV = (1 - m)^ + (- 2 - 2m')^
o 8m' = 2m <=>
m
= 0
(loai)
m
= l/2
m
= -l/2
^ ^
ihih
In i\) ff
Ket
luan:
m = —; m =-—. If
1;'•nciial'wi,.:;'^^
Bai
3. Cho (C
J:
y = f(x) = -x^ + 2mx2 - 2m +1
a. Khao sdt
s\i
bien thien v^ ve do thi tfng vdi m = 1.
b.
Chtfng
minh
r^ng (C^) luon di qua hai diem co

dinh
A, B vdi moi m.
c. Tim m de cdc tiep tuyen vdi (C
J
tai A, B vuong g6c vdi nhau.
Hifdng din giai
a. Kh3o sdt hkm so i?ng vdi m = 1. (Hoc sinh tif
giSi)
b.
Goi
A(x,);
yo) la diem codinh cua (C), khi d6: ' '
yo =-x'*+2mxo-2m+ 1; Vm , ^;
; ;);
^ /o,^
0 2m(x2
-l)-(y„+x^-l)
= 0;Vm
o
2m(xf,
- 1)= y„ +
XJ5
-1;
Vm
xf,-l
= 0
yo+x^-l
= 0
X„
=1 : •

X()
=
-1
• (1) Ub
,
E
}\ V
'= (O'k
tr v;
+
yo=l-Xo
;• ^
x„= 1 z^yo
=
0=>A(l;0)
, f
X„
=
-l
=^y„ =
0=:>B(-l;0)
.gfirMJD m,;;;!:
c.
Tiep
tuyen tai A(l;
0) c6 he so g6c
k,
=
r(l)
= -4 +

4m
Vrr'Jid .if) S.
iiiS
Tiep
tuyen tai B(-l;
0) c6 he so goc kj =
f'C-l)
= 4 - 4m . ;,,,;
oiJd:-!
?
Tiep
tuye'n tai
A, B
vuong
goc
vdi nhau <=>
k|.k2 =-1 IT
o
(-4 +
4m)(4
-
4m)
=
-1
o
3
m
=
—
4

'?
5
m
=
—
4
Bai
4.
Cho ham
so y = f(x) = x" - 2mV +
1
(1)
vdi
m la
tham
so
a.
Khao
sat ham
so (1)
khi
m = 1.
b.
Tim m de do
thi h^m
so (1) c6 ba
diem
cifc
tri 1^
ba

dinh ciia
mot
tam
giac
vuong can.
Hifdng
d§n
giai
a.
Khdo
sat ham
so (Hoc
sinh tif
giai).
b.
Ta
c6:
y' =
4x'
-
4m^x
=
4x(x^
- m^)
rx=o
y'
= 0o
do
at
X

= m
X =
-m
•T->
Dieu kien
de do
thi ham
so (1) c6 ba
diem
cifc
tri
li y' = 0 c6 ba
nghi^m
phan bi?t
o m
?i
0.
^ . f = (' i
Gpi
A(0; 1); B(m;
1
- m"); C(-m; - m'*) la cic
diem
cifc
trj.
Tac6:
AB= Vm^+m**, AC = Vm^+m**, BC =
N/W
Dieu kien
de

AABC
vuong can
la
BC^
=
AB^
+ AC^
m
= 0
(loai)
o 4m^
=
2m^
+
2m''
o
m*
-
m^
= 0
<=>
Ke'tluanims l;m
=
-l
m
= l
m
=
-l
Bai 5. Cho ham

so y = f(x) = ^ (1)
x-1
a.
Khao
sdt va
ve do
thi
(C)
cua ham so
(I).
b
Tim cac diem
thuoc
(C) c6
loa
do la
nhiTng
so nguyen.
c
Tim
diem
M e (C) de
long khoang
each
liT
M
den hai du'dng
tiem
can cua
(C)

nho
nhat.
HU<?ng
d§n
giai
•
a.
Khao
sdt va
ve do
thi
(C)
cua h^m
so <
jjiiiiodl
.v
1.
Tap
x^c
dinh:
D = R\ 1} _ ; ,
2.
Sy
bien
thien
cua ham
so
a. Gidi han, tiem can:
j' o>'
,.

2x-l , ,. ,. 2x-l
lim
y - lim +oo;
Iim
y = lim = -oo i ,. .
=>
X
=
1
la
tiem can duTng cua
do
thi ham so.
hm
y= lim y = 2
=>
y = 2 la
tiem can
ngang
cua
do
thi ham so.
b. Bang bien thien:
y' = , < 0 ; Vx e D
(x-1)^
-00
—00
+00
+00 •
Ham

so
nghich bie'n tren
cac
khoang (-oo;
1);
(1; +oo)
3.
Do
thi
(Hoc
sinh
M ve
hinh)
^J'^
''
x
=
0^y=l
'
1
y
= 0 => X = -
2
A'
i
Nh^n
xet: Do
thi
nhan
giao

diem 1(1;
2)
cua hai
di/cJng
ti^m can lam tam doi xufng.
b.
GoiM(x„;yo)6(C),Xo+l.
v
Tac6y„=
^^""^^2
+ —!—
'^O -1 X„ -1 , :.g
,.,^.,.,\
Do X(),
yo e Z
nen ta
c6 cac
triTdng
hdp sau:
igj a!?;; • , < ^
• Xo-1
=
1
c^x„ =
2=>y„
=3=>M,(2;3)
•
Xo-
1
=-1 0x0 = 0

=>yo=
1
=>M2(0; 1)
c.
Taco:
;;' Z '
<:
'
Tiem
can duTng (d,):
x -
1
= 0
Tiem
can ngang (dz):
y - 2 = 0
15
U^ngua
ae
miOc
Ic
m uii t
mien
oat;,
irung,
num
ivunrm-
lyguyen
vun i
tivng-

Goi
M
(X(,;
y„) e (C). Khi d6:
d(M;d,)=
|xo-l|
'2x0-1
. 1
d(M;d2)=
yo-2
Xo
-1
-2
Xo-1
Khi
d6,
tong khoang
c^ch
\.\S
diem
M(X();
yo) den hai
di/cfng tiem
can Ik
d
=
d(M;d,)
+
d(M,d2)
^

,1
Xo-1
+
x„-l
•dn,i„
=
2khi
XQ-I
1
Xn -1
=
2
Xn
-1
o(xo-ir=10
Xo=0
Xo-2
•
xo
=
0=>y„=
1
=>M3(0;
1)
•
Xo = 2 z:> yo = 3 =^
M4(2;
3)
2x
Aihly

Bai
6.
Cho
hamsoy
=
f(x)
=
x
+
1
a.
Khdo
sdt sif
bie'n thien
va ve do thi (C)
ciJa
ham
so'
da
cho.
b.
Tim toa do
diem
M e (C),
bie't tie'p tuyen
cua (C) tai M c^t hai
true
toa dp
Ox,
Oy tai A, B va

AOAB
c6
di?n tich bkng
_
4
Hi/dng
din
giai
a.
Khao
sdt hkm so (Hoc
sinh
tiT
giai).
b.
Gpi M
(xo;
yo) 6
(C);
y„ = f(xo)
PhiTdng trinh tie'p tuyen
cua (C) tai
M(xo; yo)
la
(d):
y =
f (xu)(x
-
XQ)
+ f(xo)

y
=
-x +
-
2x^
(Xo+1)'
(Xo+1)'
Toa
dp
giao diem
A cua (d) vk Ox
Ik: A(
-Xo
;
0)
(
2x2
Tpa
dp
giao diem
B
ciia
(d) va Oy la: B 0; ^
De thay AOAB vuong
tai O
S^QAB
=^0A.0B
1-1^2 2x2
Xn.
4

2
"(xo+1)^
4x^-(Xo+1)2=0:
2xo
+
X() +1
= 0
2x2-Xo-l
= o'
Ket luan:
M,
;M2(1;1)
Xo=
Xo=l
if
III.
CAC BAI
TOAN
TI;
LUYfN
c6
HU6NG
DAN
Bai
1.
Khao
s^t
sir bie'n thien
vk ve do thi (C) cua hkm so y = f(x) =
x^

-
3x^
+ 1.
1\i
do, bien luan theo
m so
nghiem
cua
phiTdng trinh
sau:
,
.v!'
-x^-x2+m+2=0
HiMngdSngiai
o^Ti, in, ? ,
•
m < -2
hokc
m >
—: Phi/Png trinh
c6 mot
nghiem.
2
•
m = -2
hokc
m =
—: PhiTPng trinh
c6 hai
nghiem.

1
S m
-J hod
t.
2
•
-2 < m <
—: PhiTPng trinh
c6 ba
nghipm.
f , ,
3
• u) ;.u ~ f '
Bai2.
Chohkmsoy
= f(x) = x'-mx + m- 1 (1)
a.
Chtfng minh
r^ng
tie'p tuyen
cua do thi hkm so (1) tai
diem
Xo
=
0 c6 he so
goc
nho
nhat.
b.
Vdi gia tri nko cua m do thi

cija
hkm so (1)
tiep
xuc vdi Ox.
Khao
skt vk ve
do
thi hkm so (1) gik tri tim
diTpc
cua m.
c.
Xkc
dinh
m do thi hkm
s6'(l)
c^t
true hoknh
tai 3
diem phan
bi^t.
HUdng
dSn
giai
'
a.
f
(X)
=
3x^
- m > 0 - m =

f (0)
''^^ '
=>
hp so gdc
tie'p tuyen
tai
diem
x = 0 Ik nh6
nhat.
^^'
*
b.
m = 3; m =
—
,41 li -
4
rm>0
'i'"' '
c.
x' - mx + m - 1 = 0 o (x - l)(x^ + x+ l-m) = 0=>
2
.JiifP'M'
4
k-
Bai
3.
a.
Khio
sat si/
bie'n thien

vk ve do thi
(C):
y = f(x) = x' + x - 1.
b.
Gpi
Xo
Ik
nghipm
cua
phiTPng trinh
f(x) =
0. Chtfng minh r^ng:
2Xo
-
X(,
-1<0 M
(trr
Hi^ng
din
giai
f(x) lien
tuc
tren
(
n 17 . r
V 2
=
ViEN
TIWH BINH THUAN
Luyfn

giii di
trade
kp thi DH 3
miin
Bdc,
Trung,
Nam
Todn
hoc
-
Nguyln
Van
ThOng
f(x) CO
nghiem X(,
•(xo- 1)
x„
+
—
2
<
0 =>
2xf)-X(,-1
<0 (dpcm)
Bai4.Cho(CJ: y = f(x) = mx' + (m-
l)x^+1
- 2m '
a. Tim m dc ham so' co mot diem cifc
tri.
b.

Viet
phiTcfng
trinh
tiep tuycn cua fC, ^ di qua g6'c toa do.
>f{
£ > tu »
.
2)
a. m < 0
hoac
m > 1
b.
(d,): y = 0;
(d:):
y = ^x ; (d,): '"^
Bai5.Cho(C):y
= f(x) = (x + 1)^ (x- 1)^
/.m
-
''X
•
(xYt
)
.S
'a. Khao sat va ve do thi (C) cua ham so'.
TiT
do, bien luan
theo
a so' nghiem cua
phi/dng

trinh
x"*
- 2x^ + 11 - a = 0.
b.
Tim m de parabol (P): y = mx^ - 3(m^0) tiep xuc vdi (C). ,,
Htfdng d§n giai
.^j,,,
a. • a < 10 : Phi/cfng
trinh
v6 nghiem
•
a = 10 : Phu'dng
trinh
c6 1 nghiem. _ ^ r
•
10 < a < 11 : Phu'dng
trinh
c6 4 nghiem. j ^'g;
•
a = 11 : PhiTcfng
trinh
c6 3 nghiem.
•
a > 11 : Phifdng
trinh
c6 2 nghiem.
b.
(P) tiep xuc vdi (C) khi va chi khi hoanh do tiep diem la nghiem cua he
phiTcfng
Irinh

x^-2x^
+ l =
mx2-3
4
•
m =
4x -4x=2mx
7^
-2 ft)
Bai6.Cho(C):y
= f(x)=
-x'*-3x^+
2 2
19^'-
a.
Viet
phiTdng
trinh
tiep tuyen cua (C) tai diem x =
Xo.
b.
Chufng minh rang hoanh do giao diem cua (C) va (d) la nghiem
ciia
phiTdng
trinh:
(x -
x,,)^
(xf, +
2x„x
+ 3xf, - 6) = 0

Hifdng d§n giai
a. (d): y = (2
x;',
-
6x„)x
~~
+
3x1 + x
2x
0
b
Phu'dng
irinh
hoanh do giao diem
(2x,l-6xo)x x^3xi,+-
= ^-3x +-
o(x-xo)'(x^+2x„x
+ 3x^6) = 0 x/r
:,,„•
Bai
7. Cho ham so y = r(x) = p
a. Khao sat sir bien thicn va ve do thi (C) cua ham so.
b
Tim m de di/c^ng thang (d): y = 2x + m c^t (C) tai 2 diem phan bi^t A va B.
tim
tap hdp trung diem I cua AS.
t.
m
+ A« j
8- ^ (£)^j '"" '

Hrfdng
dSn giai '(.m.utll
b
—=- = 2x + m o
<^
,
x
+
1
[2x^+(m
+
4)x + m
+
4 = 0 -f +
-fi:^r.:
Dieu
kien de (d) c^t (C) tai 2 diem phan biet la m < -4
hoSc
m > 4.
Tap hdp trung diem I cua AB la phan diTdng thang 2x + y - 4 = 0, gom
x- rn + 4 m + 4'\ .
ik>M 4
nhffngdiemM
; vdim<-4ho$cm>4. ,
:,,[•••.
v
4 2 y . — -
2x
—
1

Bai
8. Khao sat va ve do thi ham so y = l(x) = .
Tilf
do, tim m de phiTdng
x
+ 2
,
. 2sinx-l . , , - '* 'sil »tn
i
•
trinh
= 2m - 1 c6 dung 2 nghiem x x;
sin X
+ 2
HiAJngdSngiai
al
=.(x)t
mil
*
Khao sat va ve do thj (Hoc sinh
tiT
lam)
* Dat
t =
sin x;
x e
[0;
71J
^ t e
[0;

IJ n « (,.,
•^rmd^
fa; ;
PhiTdng
trinh
trd thanh: = 2m - 1 (1) " '
Dieu
kien de phiTdng
trinh
c6 dung 2 nghiem x e [0; n] la phtfdng
trinh
(1) co
dung 1 nghiem t e [0; 1).
Khao sat ham so g(t) = =^ iren [0; 1) => - < g(t) <3z:>-<m<2
KHAO
SAT
MpT SO HAM SO DA
THllfC
NANG
CAO
Bai
1. a) Bie't r^ng d6 thi ciaa hhm so'y = (3a' - l)x' - (b' + l)x' + 3c'x + 4d co
hai
diem cifc tri la:
(1;
-7), (2; -8). Hay xac dinh tong M = a" + b' + c' + d'
b)
Chu-ng minh rang do thi ham so y = x' + 2mV + 1 luon
citt
difdng thang

y
=
X
+ 1 tai
diing
hai diem phan biet vdi moi gia trj m.
I
¥unf^,
t^urri
i
utin
n\n, -
ii^ujfcri
run i
riurig
Hifdng
din giai
a) De ddn giSn, ta dat A = 3a^ - 1, B = -(b^ + 1), C = 3c^ D = 4d, h^m so da
cho chinh m y = Ax^ + Bx^ + Cx + D
Ta CO y' = 3Ax^ + 2Bx + C y' = 0 «•
3Ax^
+ 2Bx + C = 0
PhU"dng
irinh
nay c6 hai nghicm
phan
bi^t la x = 1, x = 2 nen ta c6 h? sau
f3A + 2B + C-0 ^(^)i=^=
Yo^mifii, !
r , i«a

_12A
+ 4B
+
C = 0 , „x ^
g _ ry(l) = _7 [A + B + C = -7 rniT /
Ta cung c6
y(2)
= -8'^ l8A
+
4B + 2C + D = -8 f> '
TCf
cdc dieu kien n^y, ta tim diTdc A = 2, B = -9, C = 12, D = -12 ta
tinh
diTcIc
cdc gia tri
tiTcfng
tfng la a = ±1, b = 2, c = ±2, d = -3. - ^
VayM
= a^ + b^ + c^ + d^=
1^
+ 2.2^ + 3^= 18 x^i
b)
Phi/dng
trinh
hoanh
dp giao diem cua hai do thj Ik
x"
+ 2mV + 1 =
X
+

1
o x(x' + 2m^x - 1) = 0
j.,,
De thay phifdng
trinh
nay co nghiem x = 0, ta can chifng minh phifdng
trinh
c6n lai c6, nghipm duy
nha't
khac
0.
That vay, xet ham so f(x) = x' + 2mx^ - 1.
Ta c6
f
(x)
= 3x^ + 2m^ > 0 nen ham so nay dong bien tren tap so thi/c.
Hdnnffa
lim f(x)= lim (x"^ + 2m^x -1)
=-oo
Va
lim f(x)= lim
(x''+2m^x-l)
=
+oo
Nen phi/dng
trinh
f(x) = 0 luon c6 nghi$m.
Suy ra phi/dng
trinh
f(x) = 0 c6 dung mot nghiem nay cung khdc 0 do

f(0)
= -l.Tac6dpcm. i _
Bai
2. Cho ham so y = -x^ - 3x^ + mx + 4 vdi m 1^ tham so thifc
a) Khdo sat sir bien thien va ve do thi do thi hhm so khi m = 0.
b)
Tim tat ca cac gid tri cua tham so m de h^m so da cho nghich bien tren
(0; +00).
Ht/dng
dSn giai ^
a) Vdi m = 0, ta c6 h^m so y =-x^ - 3x^ + 4.
•
Tap xdc dinh D = R.
Gidi
han cua
hjim
so la
limy
=
-oo,
limy
=
+oo
X-»+oo
X-»-00
,^
•
Chieu bia'n thien: Ta c6 y' = -3x^ - 6x,
y'
= 0 o -3x^ - 6x = 0 X = 0, x = -2. y

COngtyTNHHMl
^
L
v
vii
KHung
vict
H^m
so nghjch bien tren moi khoang
(-oo;
-2), (0;
+oo)
va dong bien tren
khoang (-2; 0).
Ham so dat ciTc dai tai x = 0 va dat ciTc tieu lai x = -2.
•
Bang bie'n thien ' • I i . ' . •
—00
-2
0
+C0
0
+00
0
•
Ta c6 y" = -6x - 6, y" = 0
o
X
= -1 nen diem uon cua
do thj la (-1;2)

Do
thi cii
true
lung tai diem
(0;
4) va cat
true
hoanh
tai
diem(l;0),(-2;()).
Do
thi cua ham so: t
b)
Ham so da cho nghich bien
tren khoang (0;
+co)
khi va
chikhi
^ , X
y'
= -3x^ - 6x + m < 0. Vx > 0
»
3x^ + 6x > m, Vx > 0
-00
Hinh
7. Do thi ham soy =
-JC'-3X^-^4
Ta
CO
bang

bien thien cua ham so g(x) = 3x^ + 6x tren (0;
+oo).
0
+00
+00
Tir
do, ta difdc dieu
kiOn
cua m la m < 0.
Bai
3. Cho ha
m
so y = f(x) = Sx'* -9x^+1 r ; . •
a) Khao sat sir bien thien va vc do thi ham so tren. - i 1 '' - •
b)
Dira vao do thi tren, hay bicn luan iheo m so nghicm cua phiri^ng
trinh
8cos'x
-
9cWx
+ m = 0 vdi X e [0;
71].
HU<tng
dan giai
a) Tap xdc dinh D= R.
Gidi
han cua ham so lim (Hx-*-9x^ +
1)
= lim (8x^-9x^ +
1)

=+oo.
X->-KC
*
Chieu bien
ihicn:
x->-a!
3
Ta c6 y' = 32x' - 18x. y' = 0 x = 0, x = ±- . ii 0 m c:^ i
Li^n
giii di tniOc Aj> thi DH 3 miin Bdc, Trung, Nam Todn hQc - NguySn Van Thdng
3
—
;+tx3
4
Ham
so dong bicn
tren
Va nghich
bi6'n
tren
Ham so dat
ciTc
dai tai (0; 1) va dat
ciTc
tieu tai
( 3^
-co;
0
; —
I 4;

4j
( 3.
49
^
r3.
49^
. 4'
32
J
.4'"
32
J
X
-00
4
0
, X
4
+00
y'
-
0 +
0
0
+
y
+00
^ 49
32
^1

49 ^
32
^ +CO
4 ' 32
^^/3 13^
Ta CO y" = 96x'- 18, y" = 0 o x = ± —, y = -— nen hai
diem
uon cua do
thi
ham so nay la
4 32
Do Ihi cua h^m so c^t true
tung
tai diem (0; 1) va c^t true hoanh tai bon
diem
phan biet la (-1; 0), (1; 0)J
•3
• Do thi ham so
b) Dat I =
cosx,
X e [(); n], ta c6
t
e[-l; 1] va ro rang moi gia trj
cua
t (t e[-l;
1
]) cung cho ta duy
nha't
mot gia tri cua x. phiTdng
trinh

da cho
ti/dng
diTdng vdi
f(t) = St' -
91^
+
1
=
1
- m
Day
chinh
la phifdng
trinh
hoanh
do
giao
diem cua hai do thi
y
= 8x''-9x'+l,y= 1-m.
Di/a
vao do thj, ta thay r^ng
49
1
;0
1
n
4
.2N/2 ,
Ncu l-m>lvl-m<-

81
'y
y
=
1
- m
(d)
1/
+
"K
3 /
\
"V
X
It
I
1
4 /
\
>.
i
1
-2 -1
1 /o
1
2 x
~ 32
-2-
32 Htnh 8. Do thi ham soy = 8x^-9x^ +
1^

o m < 0 V ni > — ihi phiTdng trinh vo nghiem.
Neu 1 - m = 1 o m = 0 ihi phiTdng trinh co diing mot nghiem.
COng ty TNHH MTV DWH Khang Vi?t
Ncu 0 < 1 - m < 1 o 0 < m < 1 Ihi phiTdng
Irinh
c6
dung
hai nghiem.
Neu
A',,
< 1 - m < 0 o 1 < m < ^ Ihi phifdng U-inh c6 dung bon nghiem.
32
32
Neu I - m = <=> m = |i ihi phi/dng trinh c6 dung hai nghiem.
32 32
Bai
4. Cho ham s6> = f(x) = mx' + 3mx' - (m - l)x - 1 vdi m la tham so'. '
a) Khao sat sir bien
thien
va ve do thi cua ham so khi m = 1
b) Xac
djnh
taft
ca cac gia tri m dcf ham
scTy
= l(x) khong c6 cifc
tri.
Hif(}ng d§n giai '
a) Vdi m = 1, ta CO ham so y =
x-*

+ 3x^-1. ^x,^
.i.'-yr
, , 5}^ ,,
Tap x^c
dinh
D = M.
Gidi
han cua ham so: > /
lim
y= lim(x-^+3x^-1) = , va jim y = lim^(x^+3x2 -1) =+«).
X->-OD
X->-00
* Sy bien
thien
ciia ham so:
2
x->+oo
Ta CO y' = 3x + 6x, y' = 0 o x = 0, x = -2.
Ham
so nay dong bien
tren
cac khoang (0;
+00),
(-co;
-2) va nghjch bien
tren
(-2; 0).
* Bang bien
thien:
Ta CO y" = 6x + 6, y" = 0 o X =-1

nen
diem uon cua do thi la (-1; 1)
Do thi ham so cat true tung tai diem
(-1; 0) va cat true hoanh tai ba diem '
phan biet.
* Do thi ham so: '
h) Ta xel cac trU'ctng hdp sa
- Khi m = 0 thi y = X - 1 ''^
nen ham so khong CO cifc trj.
- Vdi m ^ 0 thi y' = 3mx' + 6mx - m + 1.
Him so nay khong c6 cifc trj khi v4
chl
khi phiTdng
Irinh
y' = 0 khong c6
nghiem hoSc c6 nghiem kep, tiirc la
A'< 0 «. 9m^ + 3m(m - 1)= 12m^ - 3m < 0
<=>
0 < m < ^.
Luy(n
giii di
trade
kj>
thi DH 3
miSn
Bdc,
Trung,
Nam
ToOn
hoc -

NguySn
Van
ThOng
VSy
dieu kien can nm la 0 < m <
^
't> 0 > m > • •
B&i5.Chohamsoy = -2x' + 6x^-5c6d6lhi(C)
a) Khaosatv^
vedo
Ihihamso (C) '• • , Vl
b)
Vie't
phiTdng
trinh
ticp tuye'n cua (C) di qua diem
A(-l;-13).
ji:
• Hi^ng d§n giai
a) Hoc sinh
tiT
khao
sat. ,v.,
b)
Ta
CO
y' = -6x' + 12x. Goi
M(x„;
y„) \k tiep diem cua (C) vdi tiep tuy^n can tim.
Khi

do
y,i
=
-2xi'',
+ 6xf, -5 . _ ^ ,,, „,,
Phifdng
trinh
tiep tuycn cua (C) tai M y -
y,,
=
f
(X())(x
- x„) hay
y
= (-6xf) +
12x,)
)(x - x„) -
2xf)
+ 6xo - 5
Tiep tuyen nay di qua diem
A(-l;
-13) nen
-13 = (-6x,2, + 12xo)(-l - x„) -2xf, + 6x?, -5
I'Vi
o-13 = -2xo+6xo-5 +
6xo+6xo-12x„-12xo
o xf, - 3X()
+
2 = 0 o X(, =
1V

x„ = -2
Ta
CO
cdc tung do tiTcJng iJng la y(l) =-1, y(-2) = 35.
-
VdiM(l;-l)
thi phiTdng
trinh
tiep tuyen can timia:
y+l=6(x-l)c^y
= 6x-7
-
Vdi M(-2; 35) thi phu'dng
trinh
tiep tuyen can tim Ik '
y
- 35 =-48(x + 2) o y =-48x - 61
Bai
6. Cho ham so y = x' - 3x^ - 9x + m vdi tham so m.
a) Khao sat sif bic'n thien va ve do thj hkm so da cho khi m = 0
b)
Tim tat ca cac gid tri m de do thi ham so c^t true hoanh tai 3 diem phan biet
c6 hoanh dp lap thanh cap so cpng.
Hifdng dSn giai
a) Hoc sinh
tiT
khao
sit.
b)
Do thi him so da cho c^t true hoanh tai ba diem phan biet c6 hoknh do lap

thinh
cap
so'cpng
khi va chi khi phiTdng
trinh
sau c6 ba nghi^m phan biet lap
thanh cap so cpng:
x"^
- 3x^ - 9x + m = 0 (*)
Theo djnh li
Vi-et
thi tong ba nghi^m cua phU'dng
trinh
(*) li 3. Do d6,
theo
tinh
chat
cua tong cua cap so cpng thi 1 chinh la nghi$m cua phu'dng
trinh
(*)
hay
l'-
3.1'- 9.1 + m =
Oom=ll.
Vdi
m =
11,
phiTdng
trinh
(•) trd thanh

x'-
3x^ - 9x + 11 = 0 o (X -
l)(x^
- 2x - 11) = 0
Ceng
ty
TNHH
MTV DWH Khang Vi(t
Phildng
tiinh
nay c6 ba nghi^m phan bi^t
Ik
1
- 2>/3 ,1,1 + 2%/3 thoa man de bai.
Vay
gia tri can tim cua m la m = 11.
Bai
7. Cho hkm so y = f(x) = x' - 2x^ c6 do thj (C). f"" ' '
<•
'
a) Khao sat va ve do thj (C).
'i ,i, r
t,j t-f v j ! '"^
b)
Tren do thi (C) lay hai diem phan bipt li A vi B c6 hoinh dp Ian
li/dt
\h a, b.
Tim
dieu
kipn

cija a, b de tiep tuye'n ciia (C) tai cAc diem A va B
song song
vdi
nhau.
Htf^ngdSngiai
mm so y = f(x) x" - 2x^ c6 tap xac djnh D = R.
Gidi
han cua ham so: lim y =
+oo
; Um y =
+oo
i
S\ibien
thien cua ham so': N d
-uyr
« i x .
Tac6y' = 4x'-4x =
4x(x^-l),y'
= Oox = 0,x = +l. " ^
Ham so dong bie'n tren
(-1;
0),
(1;
+oo)
va nghjch bien tren
(-oo;
-1), (0; 1)
Diem
cifc dai ciia do thi Ik (0; 0), diem cifc tieu cua do thi ham so' la
(-1;

-1),
(1;-1).
Bang bie'n thien
iftj'
'*
-00
'
-1
+00
0
0
+00
+00
Ta c6 y" = 12x^ - 4 = 4(3x^ - 1), y" = 0 « x = ±
Cac diem uon cua do thi la
f
S 5^
(J3
5]
I
3' ^
[3- 9J
*
Do thi cua ham so'
b)
Ta CO
f
(X)
= 4x^ - 4x.
Gpi

a, b Ian
li/dt
la hoanh dp cua A vk B.
Hp
so gdc cua tiep tuye'n cua -3
(C)
tai A va B Ian
li/dt
la
kA
=
f
(a) = 4a^ - 4a, ke =
f
(b)
= 4b' - 4b.
Tiep tuye'n tai A va B Ian
lUdt
c6 phiTdng
trinh
la
y
=
f
(a)(x - a) + f(a) =
f
(a)x + f(a) -
af
(a)
y

=
f(b)(x
- b) + f(b) = r(b)x + f(b) - bf(b)
Hai
tieo tuve'n n^v
song song
hokc trung nhau khi \k chi khi
Luy^n
gidi
di
truOc
thi
DH
3
mijn
Bdc.
Trung.
Nam
Todn
hoc
-
NguySn
Van
Thdne
ICA
=
ke
«
4a'
-

4a
=
4b'
-
4b c:>(a
-
b)(a'
+
ab
+
b'
-
1)
= 0
Do
A va B
phan bict
ncn
a *
b, suy ra
a' +
ab
+ b' = 1
M3l
khic,
hai
tiep tuyen cija
(C)
tai
A va B

irung nhau khi
va
chi
khi
^ .
.j.
a-
+ ab +
b^
= l,a ^ b ^ a^ + ab +
b^
= l,a ;t b ( h%
;
^j,.
^)|,;f>{
f(a) -af'(a)
=
l"(b)
-
bl"'(b)
[-3a^
+
2a^
=
-Bb"*
+
2b^
\
•<!]
hh

it;nT
,
Giai
he nay, ta
diTdc
hai
nghiem
la
(a;
b) =
(-1; 1), (a;
b) =
(1;
-1).
Vay dieu kien
can
va
du
dc
hai
tiep tuyen ciia
(C)
tai
A va B
song song
vdi
nhau
la
a"^
+

ab
+
b^
=
1,
a
9t+1,
a b.
KHAO
SAT MQT SO HAM SO
PHAN
THaC
NANG CAO
Bai
1.
BicTt rang
del thi ham so'
y = ^ , ac ^
0. ad
- be ^ 0
c6 tam doi
xtfng
la
r
2;-
2
va
di qua
goc
toa do. Xac

djnh tung
dp cua
diem
c6
hoanh
dp
la
thupc
do
thi.
Hiidng
d§n
giai
; '
Do
do
thj
CO
tam
doi
xufng
la
I
nen
2c
+
d
=
0,
- = -

d
=
-2c,
c
=
2a
^
^
2
2
Do thi
di qua goc toa do
ncn
b
=
0.
Do do y(
1)
=
^
^
^
= ^ = -—
^ c +
d
2a + (-4a)
2
Vay tung
dp can
tim

la .
Bai
2.
a)
ChiJng minh rang
vdi
moi
m ^ 1
thi
do
thi cua ham
so y =
^?^^^—^——?^
luon
tiep xiic
vdi
diTcfng phan giiic
cua
goc
phan
tu"
thi? nha'l.
^
K\ - ' 1* +(m
+
2)x
+
2m
+ 2
0) lim

m de
ticm
can
xien
cua
do
thi ham
so y =
tiep
x
+
2
xuc
vdi
dirdng cong
(C):
y = x' -
3x'
-
8x.
i'
Hifdng
dSn
gial
a)
Ta
can
churng minh
r^ng
y =

—^^^"^
luon tiep
xuc vdi
y = x
vdi moi
X
-1
m
?t 1.
Dieu kien
de
hai diTdng
nay
tiep
xuc
nhau
la he
phiTPng
trinh
sau
c6
nghicm
Ceng
ty
TNHH
MTVDWH
Khang
Vi?,
(2m^-l)x-m
_^

[(2m-l)x-m^
=x^-x ](x-m)^=()
(m-ir
^
x-l
=
±(m-l)
x-l
=
±(m-l)
De thay
he
nay
luon
c6
nghiem
x = m
1 n6n
ta c6
dilu
phai chiJng minh.
(x
+
2)(x
+
m) + 2
2 . , ^
u\l
CO
y = x +

m
+ nen
tiem
can
xien
cua do thi
"^'•^ x
+
2
x
+
2
ham
so da
cho
chinh
la y = x +
m.
du'dng thang
nay
tiep
xuc vdi
y = x' - 3x'
_ 8x
khi
va
chi
khi
he
sau

co
nghiem:
X
+
m
=
x-^-3x^-8x
l
=
3x^-6x-8
m
=
x-^
-3x^ -9x
x^
-2x-3
= 0
o
m^x"*
-3x^ -9x
X
=
-1V
X =
3
Vdi
X =
-1,
ta CO m
=

5
va
vdi
X =
3
Ihi
m
=
-27.
^ '
Vay
CO
hai
gia
tri
m can
tim
la m
=
5, m
=
-27
j,-
i)Oi
;
Bai
3.
a)
Cho ham
so y = ^

'^^^
^
^
trong
do
p ^
0. p^
+
q^
= 1. (1)
Tim
la't
ca cac gia tn p,
q
sao cho
khoang
each
giiya
hai
diem cifc tri
la
VlO
.
b)
ChiJng minh rang
vdi moi
m
thi
do
thj cua ham

so y =
X
+(m
+
l)x
+
m
+ l
X
+
1
'
luon
CO
hai
diem
ciTc
trj
va
khoang
each
giila chung khong
doi.
c)
Khao
sat
va ve do
thj ham so
(1)
khi

p =
1,
q = 2 '
Hifdng
dan
giai
a) Ta CO y'
=
(2x
+
p)(x^ +1)
-2x(x^
+
px
+ q) _
-px^-2(q-
I)x
+ p
(x^
+
l)'
(x^
+1)^
Do
do,
dieu kien
dc do
thj ham
so
nay c6

hai diem
ciTc
tri
\k
phi/dng
trinh
sau
CO
hai nghiem phan biet
px'
+
2(q
-
l)x
- p = 0 (*).
Dieu
nay
liTPng difdng
vdi
A' >
0,
p 0 o (q -
1
)^
+
p^
> 0.
Vi
p ^
nen do thj

luon
cd hai
diem
ciTc
tri.
Gpi X,, X2
Ian
liTdl
la
hai
nghiem
cua
phiTdng
trinh
(•),
day
cung chinh
la cac
cifc tri
cua ham
so
da cho.
T
' , « • , .V
2X|
+p
2x2
+P
1
a

tinh
diTdc
gia
tri tifdng iJng
la ' ~
2x,
2x.
Tir
do suy ra
khoang
each
giffa
hai
diem cifc trj chinh
la
d^=(x,-X2)^
+
2X|+p 2x2+p
2x,
2X2
=(x,-X2r
+
2x,
2x2
Luy(n
gidi
dS
trudc
thi DH 3
miin

Bdc.
Trung,
Nam
Todn
hoc
-
Nguyin
Van
Thdng
=
(xi -xjr
V
4xfx^
J
Theo dinh li Vi-et cho phuTdng trinh (*) thi x, + Xj =
-^^3_i2
^
XjXj
= -1
P
Dod6 10 =
((x, +X2)^-4x1X2)
1
+
4xfx2^
4(q-ir
+
4
1
+

Theo gia ihiet + = 1 o = 1 _ q^, ihay vao ding
thuTc
iren. ta diTdc
1
+
-
Vai
:
10
=
((q-l)^+l-q^)
10(l-q^)
= (2-2q)(l-q^+4)oq-%4q^-5q = 0
Giai
phifdng
trinh
nay, ta thu diTdc nghiem thoa man de bai la q = 0.
Thuf
lai
ta tha'y thoa. Vay cac gia tri can tim la (p; q) =
(1;
0), (p; q) =
(-1;
0).
b)
Ta CO y = x + m +
x + 1
y'=l
1
(x

+ iy
,
y'= 0 o (X + 1)-= 1 o X = 0, X = 2.
Do
do, hai diem cifc trj cua do thi chinh la (0; m +
1),
(-2; m - 3) nen khoang
each
giffa
hai diem nay la d =
V(0
+2)^ + (m +1-m + 3)^ -2y/5 khong doi.
Ta
CO
diem phai chiJng minh. |p +
»:}
+^x>yC (l
^ ^xVo + xS)
c)
Hoc sinh tirkhao sat
Bai 4.
Cho
hamso
y =
c6
do thi (C).
4(x-3)
a) Khao sdt va vc do thi cua ham so da cho. „ ,.
,,v.
b)

Tim tpa do diem M
thupc
(C) sao cho tic'p tuycn cua (C) tai M c^t hai true
tpa do Ox, Oy Ian
liftJt
tai hai diem A, B
v&
dien
tich
tam
giac
OAB
\h

8
c)
Tim nhurng diem u-en (C) co khoting
each
den true hoknh gap 3 Ian den true tung.
Hif(}ng dSn giai
a) Tap xac
dinh
D =
(-00;
3) u
(3;+00).
. ,. ,, , ,^
*
Sirbienthiencuahamso: ,
Ta CO y' =

•
-3
4(x-3)^
<0
Nen ham so da cho nghich bien tren
tiifng
khoang xdc dinh.
Ham so da cho khong c6 cifc tn. , -v^
»
v> < -
i'-
a
.i.j
,
ac
dir&ng
tiem can: Ta c6 pB t ar 6. r
>
uv M . fr.
<
; T
Urn
y = hm
lim
= lim
'
X ^
4(x-3)J
^
1

-
-00,
lim y = lim
'
X ^
4(x-3)
=
+00
4(x-3)j
D6
thi
CO
X = 3 la ti?m can dilng va y = j la ti?m can ngang, tarn doi xuTng
la
I 3;- .
V
4j
*
Bang bien thien
.
1>X> !• O
S2:jx
m
Bii-J,nh
ill :m ,1 > A
<>r
•
X
-00
1

;
+00
y'
-
-
y
1
+00
1
4
1
—00
4
*
Do thi ham so c^t true tung va true hoinh tai (0; 0). (ve h\nh)
b)
Goi M(X(,; y„) la mot diem
thuoc
do
thi. Khi
do. tiep tuyen cua do thi tai M \k
K
-3x .
4xf)-9X()
_,
y
=
-3
-(X-X(,)
+

4(xo-3)
4(Xo-3)'
4(xo-3)'
=
y
4(Xo-3)
Giao diem cua tiep tuyen vdi true tung c6 tpa dp la
fo.4xf,-9x„
'4(Xo-3)^j
,
giao
diem cua tiep tuyen vdi true ho^nh c6 tpa dp li
f4xS-9x„.
;0
ft
lit.
Do
d6, tir dieu ki^n de
b^i,
ta c6 ' " ^
1
(4xf)-9x„)^_3
2
.2_a
_3)2 j
Gidi
phirpng
trinh
nky, ta thu diTpc 3 nghicm
1^

x„ = ^, x,, =
^(l
± ^)
I
2 4
Ttr
d6 ta xac djnh diTpc cic diem thoa man de b^i.
Bii
5. Cho h^m so y = c6 do thi (C). ,
A)
Khao sdt vk ve do thj ham so.
29
b) Vdi moi diem M ba't ki thuoc do thi (C), tim gia tri nho nha't cua tong khoang
each
tCr
M den hai true toa do. , >, j, n ;, ^
t,tij
HU
(.,•
t,\m-'
c) Tim nhOrng diem tren (C)
CO
toa do la cap so nguycn.
,•>',,,
; '.],.,.'}
HMng
dSn giai \
a) Hoc sinh
tU"
khao sat. ' ' *

(*,,**'
b)
Theo
de bai, ta can tim gia tri nho nha't cua bieu
thiJc
sau \
x-1
til
A =
+
2x + 3
3
vf;"'!'?'-
f':'~; mil \
5"
Ta tha'y rang vdi x = 1 thi A = 1 nen ta chi can x6t cac gia trj cua x
khac
i
sao cho A < 1, turc la chi can xet
|x|
< 1 o -1 < x < 1. Khi do, do 2x + 3 > 0
nen ta chi can xet 2 tru'dng hdp sau:
Ne'u-1 <x<Othi A = -x +
-X
-2x^-4x
+ l
2x + 3
2x + 3
=
f(x).

Ta CO l'(x)-— < Q nen day la ham nghich bien, suy ra
(2x + 3r
f(x)>lXO)=
i.
Ne'uO<x<lthi A = x +
i^=^^'+^^
+ ^=g(x)
2x + 3
2x + 3
rj,
. „ , 4(x^ +3x + l)
! '^'^
^^""^""T;
h^"^
^'i^'"'
suy ra
(•^x
+ J)
on
g(x)>g(0)
= i.
1
Vay gia tri nho nha't can tim la -, dat diTdc tai x = 0
1
Bai 6, Cho ham so y = 2x -1 +
•
^ x-1
a) Khao sat va ve do thi ham so'. ,
b) Tim toa do diem M ba't ki thuoc do thi sao cho tong khoang
each

tit M den
hai du'dng tiem can nho nha't.
Hifdng d§n giai
,(iOttteriq;iliji3!i%
a) Hoc sinh tuf giai. '
• " YiT .
b) Ta tha'y tiem can duTng cua do thi la x = 1 va ti?m can xien la x = 2x - 1.'
X6t mot diem M nkm tren do thi c6 toa dp la M
X();2X(,-1
+
Xn
-1
, X„ I
oong
ly
iisnti
MIV
uvVH'Khang Vii
Tong khoang
each
tiT
M tdi hai
diTcJng
tiem can ciia (C) la
d = |xo-l
2xn-
2x„-l
+
1
X,)

-1
x» - I
1
!'
N/5
Xo-1
Ta can tim gia trj nho
nha't
cua bieu thuTc nay iJng vdi moi gia tri x,, ^ 1.
2 !
Taco
d>2
Xa
-1
1
x„-l
2
Suy ra gia u-i nho
nha't
do chinh la d = -^^, dat diTdc khi
U(j ',,1'
x„-l
x„-l
<=>
Xn =
1
±
Vay d nho nhal khi M =
1+ 1 ;1+ 2
Hoac

M =
l-4=;l-i-^
Bai 7. Cho ham so y =
(x-l)-^+a
+ l
gnu
J V
X
1^'
a) Tim cdc gia tri cua a de do thi cua ham so c6 ba
ciTc
tri va chrfng minh
rSng
vdi
cac gia tri do thi cac
cifc
tri nay se n^m tren mot parabol c6
dinh.
b) Chtfng minh rang vdi moi a ^ K, do thi cua ham so y = luon cd ba
X
+X + 1
diem uon thdng hang.
HiTdng d§n giai
a) Ta cd y = x^ - 3x + 3 + -
•
y' = 2x - 3
2x^-3x^
- a
x^
Ta cd do thi cua ham

^g""""
""^"^
^^f"''
y
= 2x^ - 3x^ nhi/
dirdi
day: ;^ ^[~7
DiTa
theo
do thj cua ham so nay thi ta tha'y i. /,
rhng
cac gid tri a can tim la -1 < a < 0. /
Vdi
dieu kien tren,
theo
cong
thuTc ve toa '
dp
edc diem
ciTc
trj thi de tha'y cac
ciTc
tri
ciia do thi ham so da cho se
nlim
tren
-3
parabol y = 3x^ - 6x + 3 c^
dinh.
, W^*t,

at>n
mfo ^tuMA x -
(x^+x + l)-(x + a)(2x + l) _ x^ + 2ax + a -1
b) Ta CO y' = :i 5 — 5 5—
(x^+x + ir f (x^+x + ir
' • 2(xU3ax2+3(a-l)x-l / ^ ,
^ (x2+x) + l)'
Ta se chtfng minh r^ng phi/dng
trinh
tiTdng tfng 1^:
'ij^' i^'i (fi''
i'
>»
i
x' + 3ax' + 3(a-l)x-1 =0 (•) i
j,"^^
',1 . , , ^
~ j; jj 1
CO
dung ba nghiem phan biet. , cv' il-
,,/i'lV
Datf(x)
= x' + 3ax^ + 3(a- l)x- l,x€ R.
Ta
CO
f(0) = -1 < 0, f(-l) = 1 > 0, lim f(x) =
-oo,
lim f(x) =
+oo
dong

thdi
h^m so nay lien tuc tren tap so thifc nen phiTdng
trinh
f(x) = 0 c6 ba
nghiem phan biet thuoc cac khoang
(-oo;
-1), (-1; 0), (0;
+oo).
Do do, phi/dng
trinh
(•) c6 ba nghi?m
phSn
biet hay do thi da cho c6 ba diem
uon.
G'd sur ho^nh do cua mot trong cac diem uon cua do thi h^m so da cho 1^ x„
va day
cung
la nghiem cua phiTcfng
trinh
(*) hay
XQ
+ 3axo + 3(a -
1)X(,
-1 = 0;
khi
d6, tung do trfdng uTng cua diem
n^y
chinh 1^ yo = —5-^ . Ta se tim
Xo + Xo +1
mot quan he tuyen

tinh
giffa
Xo,
yo.
Tir
dieu ki^n cua
Xo,
ta thafy r^ng
XQ
+
3axf,
+
3ax„
+ 3a -1 = 3x„ + 3a o (x,) + 3a - l)(xo + x,, +1) =
3(XQ
+ a). '
2 :d
S„y ra y„ - + x„ +1) ^ XQ + 3a -1
^
3(Xo + x„ +1) .3
Do d6, cdc diem uon cua do thi
cung
thoa man quan h$ tuyen
tinh
nhif
tren,
ttfc
Ik chung thing hang. Du'dng thing di qua cdc diem uon tifdng tfng chinh
x + 3a-l
la

y = .
S
.+r , r
MQT S6
DANG
TONG
H0P
TOAN
LIEN
QUAN
DEN
^
DO TH| HAM S6
NANG
CAP '
Bk\. Cho (C„) c6 phifdng
trinh
y = x' + (m - l)x' - (m + 3)x - I.
a) Khao sdt ve do
thi
(C) ciia hkm so khi m = 1.
b)
Chtfng
minh r^ng vdi mpi m, hkm so' c6 ci/c d^ai, ci/c
ti€u.
Vie't phifdng
trinh
dirdng
thing di qua cic diem cifc dai ciJc tieu cua do thi.
c) Tim nhffng cap

dilm
nguyen
tren (C) do'i xtfng vdi nhau qua diTdng thing
y
=
X
v£l khong nKm tren di/dng thing d6.
J)
Tim
tham
so Ihifc m dc tpa dp cifc dai va
ciTc
lieu
cua
(Cm)
nam ve hai
phia
cua
dirdng
thing
3x + 4y - 7 = 0. -
Hufdng
d§n
giai
ii
; v ,
a) Hoc
sinh
tif
giai

m :uiT>
•:i)n
(<i^
:•
j '<.
b) Ta
CO
y' = 3x^ + 2(m - l)x - (m + 3). Phi/dng
trinh
y' = 0 cd
39
>0
A' = (m - 1)^ + 3(m + 3) =
m^
+ m + 10 =
nan
luon
cd 2 nghiem
phan
biet.
Suy ra vdi mpi m, ham so c6
ciTc
dai, ci/c
ticu.
Thi;c
hicn
phep
chia
y cho y', ta cd.
y

= x' + (m- l)x'-(m + 3)x- 1 =(3x' + 2(m- l)x-(m + 3))
—+
•
3
m-1
2(m-l)^
2(m + 3)
x
+
-
m^+2m-12
Do ycD =
ycT =
2(m-l)^
2(m + 3).
9
3
2(m
-1)^ 2(m + 3)
XcD+-
m^+2m-12
:ntfd
1'
XcT +
m^
+2m-12
Suy
ra cac diem cifc dai va
ciTc
tieu n^m tren di/dng thing cd phifdng

trinh
y
=
^ 2(m-l)^ 2(m +
3)1^ ^
m^+2m-12
va
do chinh la phUdng
trinh
diTdng
thing
di qua hai diem
ciTc
tri.
c) Ncu diem A cd tpa dp lii (x; y) thi diem doi xtfug cua A qua diTdng
thing
y
= X cd tpa dp la (x; y). Vi the ycu cau cua bai toan tiTdng diTdng vdi
viec
tim
nghiem
nguyen
(x; y) vdi x^y cua he phi/dng
trinh
I
[x
=
y"^-4y-l
Trir
hai phi/dng

irinh
ve thco ve, sau do chia hai ve' cho x - y
;>t
0, ta dU'dc
X'
+ xy + y^ = 3.
Dc
dang
tim dU'dc nghiem
nguyen
vdi x^y cua phu'dng
trinh
x' + xy + y^ = 3
ia(2;-l),(-l;2),(-2;
l),(l;-2). "^'^
Thur
lai vko he, ta nhan bp nghiem (2;-1). (-1; 2).' ''
Vay la tim dirpc cap diem
nguyen
duy nha't doi xuTng vdi nhau qua di/dng
thing
y = X va khong nam tren di/dng thing do la (2; -1), (-1; 2). ,

Lu^n
gUu di
trUOC
thi DH 3 m,.'n /U /' \.,:m / >i h \
.,•7,
',,;/(
Thdng

Bai
2. Cho ho diTcfng cong (C
J:
y =
-x^
+ mx -
x-m
a) Khao sat ve do thj
(C)
cua ham so khi m = 1.
b) Xac dinh m de ham so c6 ciTc dai, ci/c tieu.
Vie't
phiTctng trinh diTdng thing di
qua cac diem ci/c dai va cifc tieu cua do thj ham so'.
c) Tim cac diem trong mat phlng sao cho c6 dung hai di/cfng cua ho
(CJ
di qua.
Hifdng
dSn
giai
—x^
+ X
—
1
• 1
a) Khi m = 1, ham so trd thanh y = -x -
x-1
' x-1
Mien
xac djnhD=

R\{1}.
1
x(2-x) '^^te U
Taco
y =-1 + =
—^
y'
= 0khix =
0hoacx
= 2. ^, j
H^m
so
tang
tren (0; 1) va
(1;
2), giam tren
(-00;
0) va (2;
+00).
Ham so dat ciTc tieu bing 1 khi x = 0 v^ dat ciTc dai bllng -3 khi x = 2.
•
Gidi
han: Um y = lim
X->-oo
X-+-00
-X

1
lim
y = lim

x->l-'
-X
-
x-lj
x-1
=
+00;
lim y = lim
X-++00 X->+00
-X
-
1
=
-00;
lim y = lim
x-*i~
x->r
x-lj
-
-00:
-X
-
x-1
=
+00
H^m
so
CO
tiem can xien y = -x va ti^m can dtfng x = 1.
Bang bie'n thien

—00
0
1
+00
0
0
+00-
r+00
-00
-00
D6
thi ham so: (hoc sinh tif ve Hnh)
22
n fn»t
b) Ham so' y = ^"^^-m ^, ^^^^ ^.^ x(2m-x)
so y =
x-m
2
-X
+ mx - m
x-m
(x-m^
c6
ciTc dai v^ ciTc tieu khi vS chi khi m^O.
Khi
66 hai diem cifc tri c6 Ipa dp ti/dng ufng la (0; m) va (2m, -3m). Suy ra
dudng thing di qua hai diem cifc tri c6 phiTdng trinh li y = m - 2x.
c) Gia sur
(x,),
y„) la mot diem trong mSt phing ma c6 dung hai diTdng cong (CJ

di
qua. Khi do phifdng trinh y„ = '"^o ~ "^^
Xo-m
(1)
CdngtyTNHH
MTVl)\>
// . , Vi(t_
c6 diing hai nghiem
phan
bipt (an so la m).
Viet
lai (1)
diTdi
dang
m^ - (x,, + y„)m +
X(,(x,i
+ y„) = 0, ta suy ra tat ca nhuTng
diem
(xo;
yo)
thoa
man yeu cau bai toan la nhCTng diem c6 toa do
thoa
man
(x,) + y())(yo - 3x()) > 0.
Bai
3. Cho ham so y = x' - 3(m +
l)x^
+ 2(m^ + 4m + l)x - 4m(m + 1). (CJ
a) ChiJng minh rang

(C^)
luon di qua mot diem co djnh khi m thay doi.
b) Tim m sao cho
(Cm)
cit
true
hoanh
tai 3 diem
phan
biet.
c) Khao sat ve ve do thj ham so' khi m = 1
HiMng
dSn
giai
* ^ ^ x (>: nM si ui
il
/ f
a) Gia suTdiem co djnh la (x„; y„). Khi do ta c6 f"^' tJ'' f^
yo = - 3(m +
l)xf)
+ 2(m^ + 4m + l)x„ - 4m(m + 1) diing vdi moi m.
Vict
dang
thuTc tren
nhu'da
thiJc
theo
m, ta difdc
(2X()
- 4)m^ + (-3

xf,
+
8x0
- 4)m + xf, - 3XQ + 2xo -
y,,
= 0 vdi mpi m.
'2xo-4 = 0 '
Dieu
nay xay ra khi va chi khi -3xf) + 8x„ -4 = 0
xf,-3xf,+2xo-yo
=0
Tur
day giai ra dU'pc
Xo
= 2,
yo
= 0.
Vay
(Cm)
luon di qua diem M(2; 0)
CO
djnh. niiid
t»)
(Cm)
cit
true
hoanh
tai 3 diem
phan
biet khi va chi khi phtfdng trinh

x'-3(m+l)x^
+ 2(m^ + 4m+l)x-4m(m+1) = 0 (1)
CO
3 nghiem
phan
biet. TiT cau a) ta thay ring x = 2 la nghiem cua phiTdng
trinh
tren. Nhd do, bie'n doi da thufc ve
trai,
ta diTdc phiTdng trinh tiTdng diTdng
(x
- 2)(x^ - (3m + l)x + 2(m^ + m)) = 0
Tilf
day ta thay ring (1) co 3 nghiem
phan
biet khi va chi khi phi/dng trinh (2)
sau day
CO
2 nghiem
phan
bietkhac
2: ^,
-
(3m + l)x + 2(m^ + m) = 0
(2)
Dieu
nay xay ra khi va chi khi
|A
=
(3m

+ l)^-8(m^+m)>0
l2^-(3m
+ l).2 +
2(m2+m)^0
Giaira ta diTdc m 1.
Vay
vdi m
;t
1 thi
(Cn,)
cit
true
hoSnh
tai 3
dilm
phan
bi$t.
B^i
4. Cho ham so y = —. (C)
x-1
3) Khdo sdt sir bie'n thien v^ ve do thi (C).
^)
Tim hai diem A, B
thupc
(C) v^ doi xtfng
nhau
qua difdng thing y = x - 1-
Luyfn
giat
ae

truac
Kytnttitijmi g
Hifdng dSn giai
a) Hoc sinh tiT
khao
sal. !. if, i . f \ ^.r
t>i|V
b) Neu M(x, y)
M'(x',
y') 1^ hai diem doi
xtfng
vdi
nhau
qua du"dng
thSng
y = X - 1 Ihi ta CO . ,„.,
,,„,,,,,.,,
v=,. •
i)
-—- = -1 (do dirdng th^ng MM' vuong goc vdi diTdng thing y = x - 1).
x'-x
,
V V + Y'X + X'
ii)
Trung diem cua MM' nSm tren y = x - 1, suy ra ^ = — 1
Tir
i) va ii) ta suy ra x'= y + 1, y'= X - 1.
Gia suf M va M' la hai diem
thupc
(C) doi xufng vdi

nhau
qua diTdng
thang
y
= X -1 thi ta CO
y = -hav f
,
. x'^ (y + l)2
x-l
= y' = —— = ••'
x'-l
y + 1-1
Giai
he nay ta diTdc c5p diem
triing
nhau
M = M' c6 toa dp
1
1
va cap
diem M
72' ' V2J' ^ I sflj
U
2)
••••X'
,
.
Do chinh la nhi?ng diem A, B
can tim.
I>^•^/^l.

1.:. 2x^+(6-m)x + 4
Bai
5. Cho h^m so y = ^
mx + 2
a) Chtfng minh r^ng vdi moi gi4 trj cua m, do thi ham so luon di qua mot diem
CO
dinh duy
nha't.
Xic dinh tpa dp cua diem do.
b) Khao sat va ve do thi cua ham s6' khi m = 5.
Hifctng dan giai
Gia sur (x,,; y,,) la diem co djnh ciia do thj ham so.
L'u-
1'. '
2x,^)+(6-m)x„+4
^.
Khi
do ta CO y;, = ——^ '—il vdi moi m.
mx,,
+ 2
Suy ra
mXd.yi,
+ 2y„ = 2 xf, + (6 - m)X(, + 4 vdi mpi m.
Viet
phirong
Irinh
tren
diTdi
dang
nhj

thiJc
iheo m, ta diTdc
(xoyo
+ x,i)m + 2y() - 2 xf, - 6x1, -4 = 0 vdi mpi x
Suy ra
Xoy„
+ x„ = 0 vii 2y„ - 2 x,^, - fix,, -4 = 0. h
Giai
he nay ta diTdc nghicm duy nhal x„ = 0, y„ = 2. Tit do suy ra do thi ham
so luon di qua
dicni
co djnh duy
nha't
c6 tpa dp (0; 2).
(x-1)^
x
+ 2
'
(x-ir
pai
6. Cho ham so y =
a) Khao sat sir bicn thien va ve do thi ham so da cho.
b) Bien luan
theo
m so nghiem cua phi/dng
trinh
Hifdng d§n giai
a) Hoc sinh
tifgiai.
b) + m < 0: phiTdng

trinh
v6 nghipm
+
m = 0: phiTdng
trinh
co 1 nghiem duy
nhat;
+
0 < m < 12: phi/dng
trinh
co 2 nghiem;
+
m = 12: phiTdng
trinh
c6 3 nghiem;
+
m > 12: phiTdng
Irinh
co 4 nghiem.
=
m.
','•'•1:
'l:'.
Sd
liTdc
each
giai:
TiT do thi G d cau a), ta suy ra do thj G' cua ham so
v
= ^^~'^ nhir sau: Vdi x > -2, G' chinh la G. Vdi x < -2, G' la anh doi

x
+ 2
xiJng cua G qua
true
hoanh.
TCr
do thi G' ta cd ket qua bien luan neu tren.
Bai 7. Tim m dc phi/dng
trinh
sau cd 4 nghiem
phan
biet:
-3x-l
= mx-m
Hi/(tng d§n giai
Dap an. 1 <m<
6N/3-9.
Cdch I. Chu y phiTdng
trinh
luon cd nghiem x = 1 vdi mpi m.
Khiio
sat ham so
y
=
-
-3x -1
(x
;^ 1). Ta can tim m sao cho difdng
thang
y = m cat do

x-1
thi
tai 3 diem
phan
biet.
Cdch 2. Del thj ham so' y = 4|x|-^ -3|x|-1 g6m hai
nhanh,
nhanh
1 la d6 thi ham
so' y = -4x' + 3x - 1 vdi X < 0 va
nhanh
2 la d6 thj ham so' y = 4x- - 3x - 1
vdi
X > 0. Trong khi do y = mx - m la diTdng thdng
quay
quanh
diem
A(l;
0).
Hay
vie'l
phi/dng
trinh
lie'p tuye'n ke tiT A den
nhanh
1 cua d6 thj ham so'.
Jo.
BAI TAP TONG
H0P
LUYgN

THI
DAI HQC VA
LUYgN
HQC
SINH
GIOI
+
3x^-2
Bai
1. Cho ham so
y =
-x^
+
3x^
-
2. (C)
' ' ^
a) Khao sat sir bic'n thien va ve do thj (C). Tif do vc do thj (C) y =
b) Tim tren (C) nhffng diem ma qua do chi ke diTdc mot ticp tuyen vdi (C).
H\i6ng dSn
giai
a) Hoc sinh
tiT
khao sat.
b) Gia sur
M(x„;
y,,) la mot diem tren (C). Ta giai bai loan vict phiTOng trinh ticp
tuyen cua (C) qua M. Gia
su"
ticp tuyen (t) kc lit M den (C) tiep xiic vdi (Cj

tai
N(x,;
y,). Khi do phiTdng trinh cua (t) co dang
y-y, =(-3x|
+6xi)(x-xi)
Vi (t) di qua M ncn ta CO 6
:>
' +
yo-yi
=(-3xf
+6x,)(Xo-x,).
(1)
Ngoai ra, do N thuoc (C) nen ta
CO
^
y, =-x)' + 3x^
-2 • (2)
Nhir vay toa do tiep diem la nghiem cua he (1),
(2).
Ycu cau bai toan
o
he
(1),
(2)
v('Ji an la
(x,;
y,) co nghiem duy nhat
Thay y, tif phiTdng trinh
(2)
vao phi/dng trinh (1), ta di/dc

2x-|
-
3(x,)
+ l)xf +
6x„x,
-1 -y,) = 0
Lai thay 2xi'-3x„x?+xil-3xf+6x„x,-3xi^-0
^•
o(x,-x„)2(2x,+x,)-3) =
0 (3)
Ta thay he (1),
(2)
co nghiem duy nhat khi va chi khi
~^
=
x,,
o
x,,
-1.
Ttrdo tinh difdc y„ = 0.
Vay M(l; 0)
la
diem duy nha't tren (C) ma qua
66
co the
kc
dung mpt Uep
tuyen vdi (C).
Bai
2. Cho ham so

y = -
3x + 2. (C)
a) Khao sat
su"
bien thien va ve do thj (C).
b) Xet
3
diem A, B, C ihdng hang va thuoc (C). Gpi
A',
B', C' la giao diem cua
(C) vdi tiep tuyen cua (C) lai A, B, C. ChiJng minh rang
A',
B',
C
th^ng hang
c) Tim tren do thj (C) cac diem doi xuTng nhau qua 1(0;
2)
Hi^cJng
d§n giai
a) Hocsinhtirkhiiosal.
^ '
b) Phirdng trinh tiep tuyc'n cOa (C) tai diC'm
A(XA;
y^) c6 dang
y
=
(xx-3)(x-XA)
+ yA.
Phi/dng trinh hoanh do giao diem cua (C) va tiep tuyen co dang
(3x^

-3)(x-XA)
+ yA
=x'-3x
+
2
Thay yA
=
-3XA
+2 vao phUOng trinh, ta di/dc
(3x^
-
3)(x
-
xA)
+
X'A
-
3XA =
x-^
-
3x
<:>(X-XA)^(X
+
2XA)
= 0.
Nhi/
vay tie'p tuyc'n cua (C) tai A c^t (C) tai diem co hoanh dp
XA
(chinh
la

A)
va diem co hoanh dp
-2XA
(la diem A'), tufc la
XA
=
-2XA
TiTdng
tU"
Xg.
=-2XB,
Xc =-2xc .
(1)
Bay gicf ke't luan cua b^i toan sc dtfdc chufng minh nhd nhan xet sau:
Nhgn xet: X6t 3 diem A, B, C thuoc (C) co hoanh dp liTdng uTng la
XA,
XR,
XC.
Khi do A, B, C thang hang khi va chi khi
XA
+
XB
+
Xc
= 0.
ChuTng minh. Gi a
suT
A, B, C nam tren difdng thang co phU'dng trinh y = ax +b.
Khi do
XA,

XB,
XC
la nghiem cua phU'dng trinh
x'
-
3x
+
2
=
ax
+
b
o
x'
-
(3
+
a)x
+
(2
-
b) =
0
Ap
dung djnh li Vi-ct, ta suy ra
XA
+
XB
+
Xc

= 0.
NgiTdc lai, gia
suT
XA
+
XB
+
Xc
= 0. Viet phifdng trinh difdng thang di qua A, B
c^t (C) tai
C
thi theo phan thuan ta co
XA
+
XB
+
Xc
=
0
suy ra
Xc-
=
Xc
suy ra
C' trilng
C va co
nghla
la
A, B,
C

thang hang. Nhan xet diTdc chuTng minh.
Quay
trd lai bai
toan,
do A, B, C
th^ng hang ncn theo nhan xet,
ta co
XA
+
XB
+
Xc
= 0. Theo (1), ta co
XA-
+
XB-
+
Xc
=
-2(XA
+
XB
+
XC
) = 0.
Tiep tuc ap dung nhan xet ta suy ra A', B',
C
th^ng hang (dpcm).
Bai
3. Cho ham so'

y - — -
3x
- -
CO d6 thj (C).
2
X
a) Khao sat va ve do thj (C).
b) ChuTng minh rSng ham so co ba diem cifc tri phan biet A, B, C.
, rir;.
c) Tinh dien ti'ch tarn giac ABC.
d) Tim tam va ban kinh di/dng tron ngoai tiep tam giac ABC.
Htf(?ng
din
giai
T
' . 0 1
x-^-3x^+1
X
a)
Fa CO
y
= x
-
3 +
—
=
;
Tird6y'
=
0ox'-3x^+1 =0.

(1) •
vi,;.V
Dat
f(x) = x'
-
3x' +
1
thi f(-l) =
-3,
f(0) = 1, f(l) = -I, f(3) =
1
nen theo tinh
chat ham lien tuc, phiTdng Irinh y' =
0
c6
3
nghiem
XA,
XB,
XC
thoa man dieu
ki^n -1 <
XA
< 0 <
XB
<
1
<
Xc
< 3. Tir do suy ra dpcm.

LuySn
gUU
dS
trade
thi DH 3
miin
Bdc, Trung. Nam
ToOn
hoc -
NguySn
Van ThOng
b)
DiOn
tich
tarn gidc ABC c6 the
tinh
theo
cong thtfc
S
= ||(XA
-xeKyA
-yc)-(XA
-XcKyA
-YB)
2 2 /• . . \
Ta CO yA -yb =
^^^-3(XA-XB)-
1
1
-(XA

-Xy)
XA+XB
-3 +
1
(2)
••'ji
iih
Do
XA,
XH
Xf IJi
7>
nghiem ciia (1) nen
theo
djnh li
Vi-ct,
ta co
x.v + X|, + Xf = 3, XAX|, +
x„Xc
+ XCXA = 0 va XAXBXC = -1.
Thay vao (2), la difde yA -y,j = (XA -X^KXC + 1).
•
3
;fi,:.
!M
Tirdng
lir y^-yy = (XA-XCKXB+1)
Turdo suy ra
S-^|(XA
-X^KXA -XCKX^ -XC)

X^Xfj
+X^Xc
+X^XA -(XBXA +X^XB +X^XC)
'Dat X =
X^XB
+ x^Xc +
X^XA
, Y = XgXA + x^x^ + x\\^, ta c6
X + Y = XAX„(XA + Xu) +
XtXn(Xc
+ X,)) + XAXC(XA + Xc) =
-3XAXI
X.Y= (X^XB+X^XC + X^XAKX^X^
+X^Xi3+X^Xc)
= XAXB +X^X^ + X^X^.
+3-(X^XA
+X^XB
+X^Xc)
Siir
dung
hang
dfmg thii-c \\ x^ + x^ -
SXAX^XC
= (XA
+Xi3
+Xe)[(xA +XB +xc)^
-3(XAXB
+ x^Xc +
XCXA)
.

Ta
tinh
diTdc x^ +
Xy
+ = 3.(-i) + (3)(3- - 3.0) = 24.
Tirctng
tir x^x^ +
x'^x^'-
+ x;^x^ =
3X^X-BX^
+
+(XAXB +XBXC +XCXA)[(XAXB +XBXC +XCXA)^ -3XAXBXC(XA +XB +XC)'
Suyra XAXB+XBXJ + X^X^=3
Thay vao ta
tinh
di/ac
XY = 3 + 3 - 24 = -18.
Tir
day suy ra (X - Y)' = (X + Y)' - 4XY = 9 - 4(-18) = 81 * V ^t'-
Suy ra
|X-Y|
= 9 '
4
C6ng ty
TNHH
MTVDWH Khang Vi?t
c)
Hi^(^ng
dan. Hay tim cac
hang

so a, b, R sao cho (x^ - a)^ +
(yA
- b)^ = vdi
1
XA
la nghiem cua
(1)
va yA =
•
•-3XA

jj^i
4. Xet parabol (P) c6 phiTdng
trinh
y = xl M6t diTdng tr6n c^t (P) tai cac
diem
A, B, C, D c6 hoanh do tUdng u'ng la a, b, c, d. Chtfng minh r^ng a + b +
c + d = 0.
Hrf(?ngdSn
giai o J
<iv, ,.,,5
Gia suT
phu-dng
trinh
du^dng tron c6
dang
(x -
A)^
+ (y - B)^ = RI
Thay y = x' vao va bien ddi, ta

du"dc
phu"ctng
trinh
hoanh do giao diem cua
(P) va dirdng tron la
x*
- (2B - l)x^ - 2Ax + A^ + - R^ = 0 (1)
Theo gia thiet a, b, c, d la 4 nghiem ciia phu'dng
trinh
(1).
Ap
dung dinh li
Vi-et
cho phu'dng
trinh
bac 4, ta c6 a + b + c + d = 0.
Bai
5. Trong mat
phang
vdi he toa do
De-cac
vuong goc Oxy, cho di/dng cong
(C):
y = 2x'' - 3x^ + 2x + 1 va difdng thing (d): y = 2x - 1.
a) Chilng minh rang diTdng cong (C) va diTdng thang (d) khong cat nhau. u
b)
Tim tren (C) diem A c6 khoang
each
den (d) la nho
nhat.

HUiing,
dSn giai
a) PhiTdng
trinh
hoanh do giao diem cua (C) va (d) co
dang
2x^
- 3x^ + 2x +
1
= 2x - 1
o 2x''- 3x^ + 2 = 0 (1)
Dat
t = x^ thi ta duTdc 2t^ - 3t + 2 = 0. Phu'dng
trinh
cuoi cung nay c6:
A
= 9 - 16 < 0, do do
CO
nghiem. Tif do (1) v6 nghiem va nhu" vay ta co dpcm.
b)
Xet diem
A(X();
yo) thuoc (C). Ap dung cong thtfc
tinh
khoang
each,
ta tim
duTdc khoang
each
tiT

A den (d) b^ng d =
yo
•
2x0 + 1
2xo - 3xo + 2
75
Do 2x,'5-3xf,+2 =
2
\2
7 7
+
->
8 8
•'.'J'.
Dau bing xay ra khi va chi khi
XQ
=
—
O
X„ =±
—
4 2
^.V3-i^
2 8
Tur
day
ta tim
diTdc hai diem A
Ih
A,

Chu
y: Co the
kicm
tra lai r^ng tiep tuyen ciia (C) tai A,, A2 trung nhau va
song song
vdi (d). Sir kien nay khong ngau nhien. Co the chiJng minh diTdc
r^ng neu A(x„; y„) thuoc (C) la diem co khoang
each
den (d) la nho
nhat
thi
•Jircfng
thing qua A va
song song
vdi (d) se tiep xuc vdi (C). Day cung la mot
ttnh
cha't
co the
diing
de giai bai loan khoang
each.
.(Ou!,
11
'ii! n ':i> I . v
Luy?n
glai
d?
trwfr
i-f
tWi

lui '
misn
ttac.
imng,
nam wan
niM-
-
lyguyen
vurrrmmg
Bai
6. Cho ham so y =
x'
+3X + 3
x
+
1
a) Khao sat va ve do Ihi (C) cua ham so.
b)
Xac dinh hai diem A, B Ian
liTdl
d tren hai nhanh cua (C) sao cho AB
ngrii,
nhat.
Hifdng d§n giai 0^b+.
a) Hoc sinh
tiT
khao
sat.
b)
Xet diem

A(XA;
YA)
ihtipc nhanh phai (tfng vdi
XA
> -1) va
B(XB;
ya) thuot
iii
nhanh trai (tfng vdi
XQ
< -1).
.
r oia a
%.
tnii i
Ta CO the dat
XA
+ 1 = a v^
XB
+ 1 = -b vdi a, b > 0. Ta c6
AB'
=
(XA
-
XB)'
+ (yA + ys) =
(XA
-
XB)'
+

=
(a + b)' + (a + b+- + -)' = (a+ b)'
XA
-XB
+
XR
~
X
;
(XA
+1)(XB+I)
1
2
Jli/if!
or
2 2
+
•
a^b' ab
=
8(v^ + l).
Dau
bang
xay ra khi va chi khi a = b va 2 = o a = b = ^
a^b^
Vay
khi A-
.IT
ngan
nhat.

Bai
7. Cho ham so y -
x^
+2x + 5
1
thi
AB
(C).
x
+
1
' ' . not) A
"u*
a) Khao sat sir bien thien va ve do thi (C). '' *
b)
Dirdng thang (d): y = -x + m c^t do thj (C) tai hai diem phan bipt M, N. Tim
phu'dng
trinh
quy
tich
trung diem I cua doan MN.
HUdng dSn giai
a) Hoc sinh
tiT
khao
sat. ^ *J<
'^A'ACH
b)
Phu'dng
trinh

hoanh dp giao diem cija (d) va (C) c6
dang
x^
+2x + 5
x
+
1
=
-X + m o 2x + (3 - m)x + 5 - m
=:
0
Theo cong thiJc
tinh
tpa dp trung diem va dinh li
Vi-et,
ta co
_XM
+XN
_m-3
x,
2 4
Vi
I n^m tren diTdng thdng y = -x + m nen ta c6
L^wig
ijr
1111111
mt r uf rii Tvnang ri^
m-3 3m+ 3
y,
=-Xi +m = + m =

^'
' 4 4
Tir
day, khur m ta diTdc y, = 3x, + 3. Suy ra phiTdng
trinh
quy
tich
trung diem I
cua doan MN la di/dng th^ng c6 phi/dng
trinh
y = 3x + 3.
Bai
8. Cho hp
(C
J: y
m^x^
+1
.
Tim tren dU'dng thang y = 1 diem ma khong
CO
do thj (CJ nao di qua.
Hifdng din giai
Gpi
(a; 1) la diem ma khong c6 do thj (Cm) nao di qua, ttfc la khong ton tai m
sao cho
1
=
m^a^+1
Dieu
nay

tifdng
dU'dng vdi phu'dng
trinh
m'a' + 1 = a khong cd nghipm m
o a - 1< 0.
Vay
tat ca nhCTng diem can lim la nhffng diem cd tpa dp (a; 1) vdi a < 1.
Bai
9. Tim tat ca cac gia tri m sao cho (x + l)(x + 3)(x' + 4x + 6) > m vdi mpi x.
HiiTdng
d§n giai
X6t hhm so y = f(x) = (x + l)(x + 3)(x' + 4x + 6) thi ba't
dang
thifc da cho
dung vdi mpi x khi va chi khi m < min f(x). Dat t = x' + 4x + 4 = (x + 2)' > 0
thi
f(x) =
(X-
+ 4x + 3)(x' + 4x + 6) = (t - l)(t + 2) = t' + t - 2 = g(t).
Khao sat ham so' g(t) vdi t > 0, dc
dang
tim di/dc ming(t) = -2 .
t>()
Tiir
do suy ra min I"(x) = -2 va nhU' vay dap so cua bai toan la m < -2.
Bai
10. Bien luan
theo
m so nghiem cua phu'dng
trinh

x' - (4 + m)|x| + 5 + Im = 0.
Hrfdng
d§n giai
>
i
+
m < - ^: phi/dng
trinh
cd 2 nghiem;
+
m = -— : phu'dng
trinh
c6 3 nghiem;
+
—^ < m < -2 : phu'dng
trinh
cd 4 nghiem;
+
m = -2: phu'dng
trinh
cd 2 nghiem;
+
-2 < m < 2: phu'dng
trinh
vd nghiem;
+
m = 2 : phu'dng
trinh
cd 2 nghiem;
+

m > 2: phu'dng
trinh
cd 4 nghiem.
Htfdng
dan:
Vict
phUdng
trinh
lai du'di
dang
,
a
L-
< y •
x2-4
X
+
5
X
-2
•
= m
Luy^n
gidi
dS
truOc
thi DH 3
miSn
Bdc, Trung, Nam
Todn

hoc -
ISguySn
van i
nong
4^
+ 5
Do
thi ham so'd vc' trai difOc suy ra tiT do thi
hiim
so' y =
bang
X
-2
*
cdch
gii? nguyen
nhanh
phai (u'ng vdi x > 0) hOp \d\h doi xu'ng cua
nhanh
nay true tung.
Bai
12. Cho f(x) = x' - 3x. ChiJng minh rSng neu x > y thi f(x) - f(y) > -4. Dan
bang
xay ra khi va chi khi
nao?
'
Hi/(?ng d§n gial
*»)
"^n (.P)
^li

ft!)
OD
Khao sat do thj ham so f(x) = x' - 3x, ta thay f(x) dat
cUc
dai bkng 2 khi x =
-1
va dat ciTc
ticu
b^ng -2 khi x = 1.
-»> r'M** "i'
«• \ r«J
.0 =
3
y
•
' '
\
-3 -2/ - 1 "
V
II
1 w
1/2 3 "
/
-2
//i/i/i
15. Doth
f
ham soy
=
x' - 3x

•
Ncu y < -1 thi
i"(y)
< 2. Ta c6 2
triTimg
hdp:
+
Neu
X
< -1 thi do
l"(x)
tang tren
-oo;
-1) nen la c6 l(x) - f(y) > 0 > -4.
•+
Ne'ux>-1 thil(x)>-2dod6f(x)-r(y)>-4. ^
•
Ncu y > -1 ihi
X
> -1 va ta c6 f(x) > -2. Ta lai xct 2 triTtJng hdp:
+
Ncu y <
1
thi f(y) < 2 va ta c6
r(x)
- l(y) > -4.
+
Ncu y > 1 ihi do him so f(x) tang len
(1; +oo),
la c6 f(x) -

l"(y)
> 0 > -4
Vay
trong moi
irifdng
hdp ta dcu co f(x) - f(y) > -4 (dpcm).
PHlJdNG
TI^
BXT
PHl/dNG TRiNH
MC vA L6GARIT
Hfe
PHl/dNG
TRiNH,
Hfe BXT
PHIJCJNG TRiNH
VA
NfU L6GAWT
I
T6M TAT
LI
THUYET jjuw-'u.]
u,u,
1. phi/dng
trinh
cd ban
a
phuang tiinh mu
CO
ban a" = m(0< a ^ 1).

Neu m < 0 thi phifdng
trinh
v6 nghiem.
, • .7
.
!
Neu m > 0 thi phu'dng
trinh
co nghiem duy
nhat
x = logam.
b Phuang trinh Idgarit ca ban i
log„x
= m (0 < a ^ 1) o
X
= a'" 3"^^^^ ^''^ '
2. Mpt so phu'dng phap giai phu'dng
trinh
mu va logarit. 8'>Mi>
FhiTdng
phap
du'a vc cung cd so, phi/dng
phap
dat an phu, phifdng
phap
logarit
hoa, phu'dng
phap
sijf
dung

tinh
dong bicn, nghjch bicn cua ham so.
Chiiy:
" . • i , .
•
Cho 0 < a ?t 1. Khi d6 ^ ' ^ \
+
a"^'= a''*^'f(x) = g(x)
+,
loga f(x) = log, g(x) « f(x) = g(x) > 0. < •'
•
Cho ham so y = f(x) dong bie'n tren D va x,, e D.
Khi
do, phu'dng
trinh
l"(x)
=
f(X())
co nghiem duy
nhat
x = Xo tren D.
3.
phu'dng
trinh
mu va Idgarit.
Phrfdng phap chung
•
Khi giai cac he phi/dng
trinh
mu va logarit, ta difa vao cdc

phep
bicn doi ve
luy
thiTa, mu v^ logarit va
dilng
cac phiTdng
phap
the, phiTdng
phap
cong dai
so, phifdng
phap
dat an phu,
•
Can chu y dat dieu kien dc he phifdng
trinh
c6 nghiem, dac biet la cac bieu
thiJc n^m trong logarit. ,
4.
Bat phifdng
trinh
mu va Idgarit.
a-
Bat
phuang trinh mu ca ban.
•
Ba't phifdng
trinh
a" >m(0< a
7i

1). '
Neu m < 0 thi ba't phifdng
trinh
nghiem dung vdi moi gia tri cua x.
Neu m > 0 va a > 1 Ihi nghiem cua bat phifdng
trinh
la x >
log„m.
Neu m > 0 va a <
Ilhi
nghiem cua bat phifdng
trinh
la x < log^m. ,
•
Bat phifdng
trinh
a" < m (0 < a 1) •/ ,
Ne'u m < 0 thi ba't phifdng
Irinh
v6 nghiem. '
45
Liiy?n
gidi
di
trade
thi DH 3
miSn
Bdc.
Trung,
Nam Todn hoc -

NguySn
Van ThOng
Ne'u m > 0 va a > 1 Ihi nghicm cua bat phtfdng trinh la x < logam.
Neu m > 0 va a < 1 thi nghicm ciia bat phUdng trinh la x > iogam.
b. Bdt
phuang
trinh
Idgarit
ca ban
•
Bat phiWng trinh logaX > m (0 < a 1). !•;
Neu a > 1 thi nghiem cua ba't phU'dng trinh la x > a'". j Ji-,
j-
fj
Neu 0 < a <
1
thi nghiem cua bat phU'dng trinh la a < x < a"".
,5^,;.,^
•
Bat
phU'dng
trinh
logaX < m (0 < a ^ 1)
Ne'u a > 1 thi nghiem cua bat
phU'dng
trinh la 0 < x < a'".
Ne'u 0 < a < 1 thi nghicm cua phU'dng bat
phU'dng
trinh la x > a'".
c.

Phuang
phdp
giai
bdt
phuang
trinh
mu
vd.
logarit.
Cung nhU'phi/(Jng trinh mu va logarit, dc giai bat phi/dng trinh mu va logarit ta
cung
siif
dung
cac phifdng
phap:
di/a ve
cung
cd so, dat an phu, phiTdng
phap
logarit hoa, phiTdng
phap
sOc
dung
tinh
dong
bicn, nghjch bie'n cua ham
so,
Chii
y: Khi giai cac bat phU'dng trinh logarit, ta phai dac biet chu y dieu kien
xac dinh cua bat phifdng trinh.

II.
CAC BAI
TOAN
MINH HQA
Bai
1.
a. Cho
X,
y la hai so du'dng ihoa man x^ + y^ = 7xy. Chiirng minh r^ng
1
x + y 1 ^, ,
'og3 —^ = - (logix + logiy)
b. Cho
X,
y, z > 0;
X,
y, z 1 va M > 0; M 1.
ChuTng minh rkng ne'u x, y, z lap nen cap
so'nhan
thi
log^
M - logy M _ log^ M
logy M - log^ M logy M
Hi^i^ng
dSn giai
1
a.
Taco:
log3-^^-!^ = -log3
3 2

1, x^+y^+2xy 1 9xy
\ogT,
=^
= -log-,—-
1
=
2 ('^Ssx
+
log3 y)
b.
X,
y, z lap nen cap so
nhan
Ta c6:
log^M-logyM
log^M
logy M - logy M ~ logy M
xz = y^ => logyX + logyZ = 2
logyM^
logx M
1-
logy M
log,Mj_
logxM(-logyx)
lOgyM(lOgyZ-l)
logy M
jOgyM
-1
COng
ty

TNHH
MTVDWH Khang Vt(t
log,
M(logy z-l)
logy M (logy
Z-l)
log,
M - logy M ^ log, M
"^^^
k^y M - logy M logy M
nai 2. Giai cac phiTdng trinh:
log,(x'
+ x+l)-iog3X = 2x-x
•
,og,(x' +
X+
1) + log2(x' -
X
+ 1) = logjCx" +
X'
+ 1) + l0g2(x' - x' + 1)
3«
+ 5" = 6x + 2
HiMngdSngiai
'X'diuil
gnburirt^
•'
^
Dieu kien: x > 0. PhiTdng trinh tiTdng diTdng vdi
log3

x^+x +
1
=
2x-x^ o logj
Vx
-
N/X
>
+
3
De
thay:
I
Vx
N2
+ 3>3:
+
3
+
(x-l)'=l
>logj 3 =
1
V1(X-
1)^>0=:> log.
Vx
+ 3
+
(x-l)^>l
-
i

Ding
thuTc
xay ra khi va chi khi:
>/x
=
0
ox
= l
(Thoa
dieu kien)
Vay phiTdng innh c6 nghi$m x = 1
b. PhiTdng trinh tiTdng diTdng vdi:
l0g2
[(x^ +
X
+
l)(x2
-
X
+ 1)] =
log2
[(x^ +
X^
+ l)(x^ -
X^
+ l)
o(x^^lf-x^=(x^
+
lf-x^ox«-x^==Oo^^:J'
c- PhiTdng trinh tiTdng diTdng vdi: 3" + 5* - 6x - 2 = 0 (1)

De
thay
x = 0; x = 1 li hai nghi?m ciia phiTdng
irinh
(1).
X6t
ham so f(x) = 3' + 5' - 6x - 2 tren R- .
f
(x)
=
3Mn3
+
S'lnS
- 6 J '
f"(x)
=
3"(ln3)'
+
5x(ln5)'
> 0; Vx e R '~- i.
=>
f(x)
dong
bien tren R. .:
id.'tt
3" ^ ' '
A
Matkhdc:
LuySn
mi d?

trade
ky thiPH 5 mmBUc.
Trung,
Nam loan / tySn van
'IhO
f(()) = ln3 + ln5 - 6 = lnl5 - 6 < 0
r(l)
= 31n3 + 51n5-6>0
Suy ra ton tai duy
nhat
a e (0; 1) sao cho f (a) = 0
Bang bien thien
X
—00
0
'•'a
I
• +00
f(x)
-
-
0 +
+
f(x)
+00~.^_____
Vay
phiTdng trinh c6 hai nghiSm x = 0, x = 1.
Bai
3. Giai phu'dng trinh: logjoos = x'' - 3x^ -1
x''

+ x^ +1
Hifdng din giai
Dat
a = 4x^ + 2>0
b-x^ + x^+l>0
PhU'ctng trinh trd th^nh
\og2(m
~^
T
a . 2008*' ;
Neu a > b thi - >
1
>
b
2008"
Neu a < b thl - <
1
<
Vay
a = b =^
x""
- 3x^ -
1
= 0
2008^ "f4
M'iVil
b
2008"
- 1
=

Dat t = x^ t > 0. Phi/cfng trinh trd thanh: t^ - 3t - 1 = 0
Xet ham so f(t) =
t^
- 3t - 1
tren
[0;+00)
f
(t)
= 3t^ - 3
f'(t)
= 0 o
t
= l
t
=
-l
X6t
bang
bie'n thien
0
f(t)
f(t)
+C0
0
+00
PhiTcfng trinh chi c6 dung mot nghiem t e
(1;
2) nen ta dat
t
=

2cos(p
0<cp<7
J
J
2(4cos'(p
-
3cos(p)
= 1
<=>
cos3(() = ^
<=> cp
= ^
Vay
phiTdng trinh c6 dung hai nghiem x =
^2cos-^
\x=
2cos
—
9
Bai
4. Giai phiTdng trinh
l0g2(KW
=
2008^'""* ' '^^^ '"Sr
Hif(}ng dSn giai
Ta c6:
2008^l''"''U4l'-'''''''
>4'''''"''U 4l'-''''''U-4'''''''''U 4
sinx
+

cosx
.
1 2
:)t«t»>^*'»'>^*:i'f!,a
>sin'x + cos x = l)
=>
Phifdng trinh xac dinh vdi mpi x e R.
Dait= 2008^'^'"''l+4i'-'"''l-5>0
PhiTdng trinh (1) trd thanh , ^
'u
= 2008'-l [2008'-u + i :
t+
1-2008"
[2008'-2008" =u-t
Xet
ham so 1(1) = 2008' - t - 1, t e [0;
+oo)
r(t)
= 2008'.ln2008 - 1 > 0; Vt > 0
Bang bien thien
[2008'-u + l [2008'=t + l
u
= t
u
= t
(2)
t
0
+00
m

+
m
—
*• +°o
0 • 1
Khi
do, (2) suy ra u = t = 0
2QQg2|sinx|^44|sinx|
,.
I , I , o x = kn (k e Z)
^4|sinx| _ J_
^|cosx|
~4"
' • •
Vay
phi/cJng trinh c6 nghiem x = kn (k e Z)
^al
5. Gi^i phircjng trinh 2'" + 3'" = 2" + 3"^' + x + 1
HUdng dSn giai
Phifdng trinh
liTcfng
difdng vdi
2^'+3'"+2''=2''+'+3''^'+x + l (1) ' •
AO