com BaiTapGiaiTich12 KhaoSatHamSo TranSiTung

8/8/2019 com BaiTapGiaiTich12 KhaoSatHamSo TranSiTung

1/51

TRAN S TUNG

----& ----

BAI TAP GIAI TCH 12

ON THI TOT NGHIEP THPT & AI HOC

Nam 2009


2/51

Tran S Tung Khao sat ham so

Trang 1

1. inh ngha:Ham so f ong bien tren K ("x1, x2 K, x1 < x2 f(x1) < f(x2)Ham so f nghch bien tren K ("x1, x2 K, x1 < x2 f(x1) > f(x2)

2. ieu kien can:

Gia s f co ao ham tren khoang I.a) Neu f ong bien tren khoang I th f(x) 0, "x Ib) Neu f nghch bien tren khoang I th f(x) 0, "x I

3.ieu kien u:Gia s f co ao ham tren khoang I.a) Neu f (x) 0, "x I (f(x) = 0 tai mot so hu han iem) th f ong bien tren I.b) Neu f (x) 0, "x I (f(x) = 0 tai mot so hu han iem) th f nghch bien tren I.c) Neu f(x) = 0, "x I th f khong oi tren I.

Chu y: Neu khoang I c thay bioan hoacna khoang th f phai lien tuc tren o.

VAN E 1: Xet chieu bien thien cua ham so

e xet chieu bien thien cua ham so y = f(x), ta thc hien cac bc nh sau:

Tm tap xac nh cua ham so.

Tnh y. Tm cac iem ma tai o y= 0 hoac y khong ton tai (goi la cac iem ti han)

Lap bang xet dau y (bang bien thien). T o ket luan cac khoang ong bien, nghchbien cua ham so.

Bai 1. Xet chieu bien thien cua cac ham so sau:a) 22 4 5y x x= - + + b)

2 54 4

xy x= + - c) 2 4 3y x x= - +

d) 3 22 2y x x x= - + - e) 2(4 )( 1)y x x= - - f) 3 23 4 1y x x x= - + -

g) 4 21 2 14

y x x= - - h) 4 22 3y x x= - - + i) 4 21 1 210 10

y x x= + -

k) 2 1

5

xy

x

-=

+

l) 1

2

xy

x

-=

-

m) 111

yx

= -

-

n)22 26

2x x

yx

+ +=

+o)

13

1y x

x= - + -

-p)

24 15 93

x xy

x

- +=

CHNG ING DUNG AO HAM E KHAO SAT

VA VE O TH CUA HAM SO

I. TNH N IEU CUA HAM SO


3/51

Khao sat ham so Tran S Tung

Trang 2

Bai 2. Xet chieu bien thien cua cac ham so sau:a) 4 3 26 8 3 1y x x x= - + - - b)

2

21

4

xy

x

-=

-c)

2

21

1

x xy

x x

- +=

+ +

d)2

2 1xy

x

-= e)

2

3 2

xy

x x

=

- +

f) 3 2 2y x x= + + -

g) 2 1 3y x x= - - - h) 22y x x= - i) 22y x x= -

k) sin22 2

y x x

= - <

< < >

< < > >

D

1 20 0x x P< < <

5) e ham so 3 2y ax bx cx d = + + + co o dai khoang ong bien (nghch bien) (x1; x2) bang

d th ta thc hien cac bc sau:

Tnh y.

Tm ieu kien e ham so co khoang ong bien va nghch bien:

00

a >D

(1)


4/51


Trang 3

Bien oi 1 2x x d - = thanh2 2

1 2 1 2( ) 4x x x x d + - = (2)

S dung nh l Viet a (2) thanh phng trnh theo m.

Giai phng trnh, so vi ieu kien (1) e chon nghiem.

Bai 1. Chng minh rang cac ham so sau luon ong bien tren tng khoang xac nh (hoactap xac nh) cua no:

a) 3 5 13y x x= + + b)3

23 9 13

xy x x= - + + c)

2 12

xy

x

-=

+

d)2 2 3

1x x

yx

+ -=

+e) 3 sin(3 1)y x x= - + f)

2 2 1x mxy

x m

- -=

-

Bai 2. Chng minh rang cac ham so sau luon nghch bien tren tng khoang xac nh (hoactap xac nh) cua no:

a) 5 cot( 1)y x x= - + - b) cosy x x= - c) sin cos 2 2y x x x= - - Bai 3. Tm m e cac ham so sau luon ong bien tren tap xac nh (hoac tng khoang xac

nh) cua no:

a) 3 23 ( 2)y x mx m x m= - + + - b)3 2

2 13 2

x mxy x= - - + c) x my

x m

+=

-

d) 4mxyx m

+=

+e)

2 2 1x mxy

x m

- -=

-f)

2 22 32

x mx my

x m

- +=

-

Bai 4. Tm m e ham so:a) 3 23y x x mx m= + + + nghch bien tren mot khoang co o dai bang 1.

b) 3 21 1 2 3 13 2

y x mx mx m= - + - + nghch bien tren mot khoang co o dai bang 3.

c) 3 21 ( 1) ( 3) 43

y x m x m x= - + - + + - ong bien tren mot khoang co o dai bang 4.

Bai 5. Tm m e ham so:a)

32( 1) ( 1) 1

3x

y m x m x= + + - + + ong bien tren khoang (1; +).

b) 3 23(2 1) (12 5) 2y x m x m x= - + + + + ong bien tren khoang (2; +).

c) 4 ( 2)xy mx m

+=

+ong bien tren khoang (1; +).

d) x myx m

+=

-ong bien trong khoang (1; +).

e)2 22 3

2x mx m

yx m

- +=

-ong bien tren khoang (1; +).

f)

22 32 1

x x my x

- - += + nghch bien tren khoang

1;2

- + .


5/51


Trang 4

VAN E 3: ng dung tnh n ieu e chng minh bat ang thc e chng minh bat ang thc ta thc hien cac bc sau:

Chuyen bat ang thc ve dang f(x) > 0 (hoac b)

2 1sin tan , 0

3 3 2

x x x vi x+ > < >+

d) ( )2 21 ln 1 1x x x x+ + + +

Bai 5. Chng minh cac bat ang thc sau:a) 0tan 55 1,4> b) 01 7sin20

3 20< < c) 2 3log 3 log 4>

HD: a) 0 0 0tan 55 tan(45 10 )= + . Xet ham so1( )1

xf x

x

+=

-.

b) Xet ham so 3( ) 3 4f x x x= - .

f(x) ong bien trong khoang1 1;2 2

-

va 0

1 7, sin20 ,3 20

1 1;2 2

-

.

c) Xet ham so ( ) log ( 1)xf x x= + vi x > 1.


6/51


Trang 5

VAN E 4: Chng minh phng trnh co nghiem duy nhat

e chng minh phng trnh f(x) = g(x) (*) co nghiem duy nhat, ta thc hien cac bc sau:

Chon c nghiem x0 cua phng trnh.

Xet cac ham so y = f(x) (C1) va y = g(x) (C2). Ta can chng minh mot ham so ongbien va mot ham so nghch bien. Khi o (C1) va (C2) giao nhau tai mot iem duy nhat

co hoanh o x0. o chnh la nghiem duy nhat cua phng trnh (*).

Chu y: Neu mot trong hai ham so la ham hang y = C th ket luan tren van ung.

Bai 1. Giai cac phng trnh sau:a) 5 5x x+ - = b) 5 3 1 3 4 0x x x+ - - + =

c) 5 7 16 14x x x x+ - + + + + = d) 2 215 3 2 8x x x+ = - + +

Bai 2. Giai cac phng trnh sau:a) 5 5 51 2 3 0x x x+ + + + + = b) ln( 4) 5x x- = - c) 3 4 5x x x+ = d) 2 3 5 38x x x+ + =

Bai 3. Giai cac bat phng trnh sau:a) 3 4 51 5 7 7 5 13 7 8x x x x+ + - + - + - < b) 22 7 2 7 35x x x x x+ + + + + <

Bai 4. Giai cac he phng trnh sau:

a)

3 2

3 2

3 2

2 12 1

2 1

x y y y

y z z z

z x x x

+ = + + + = + +

+ = + +

b)

3 2

3 2

3 2

22

2

x y y y

y z z z

z x x x

= + + - = + + -

= + + -

c)tan tan

52 3

4

x y y x

x y

- = -

+ =

p d)

3 2

3 2

3 2

6 12 86 12 86 12 8

y x x

z y y

x z z

= - + = - + = - +

HD: a, b) Xet ham so 3 2( )f t t t t = + + c) Xet ham so f(t) = tant + t

d) Xet ham so 2( ) 6 12 8f t t t = - +


7/51


Trang 6

I.Khai niem cc tr cua ham so

Gia s ham sofxac nh tren tap D (D

R) va x0

D.a) x0 iem cc ai cua f neu ton tai khoang (a; b) D va x0 (a; b) sao cho

f(x) < f(x0), vi "x (a; b) \ {x0}.

Khi o f(x0) gl gia tr cc ai (cc ai) cua f.

b) x0 iem cc tieu cua f neu ton tai khoang (a; b) D va x0 (a; b) sao cho

f(x) > f(x0), vi "x (a; b) \ {x0}.

Khi o f(x0) gl gia tr cc tieu (cc tieu) cua f.

c) Neu x0 la iem cc tr cua f th iem (x0; f(x0)) gl iem cc tr cua o th ham so f.

II. ieu kien can e ham so co cc tr Neu ham so f co ao ham tai x0 va at cc tr tai iem o th f (x0) = 0.

Chu y: Ham so f ch co the at cc tr tai nhng iem ma tai o ao ham bang 0 hoackhong co ao ham.

III. ieu kien u e ham so co cc tr

1. nh l 1: Gia s ham so f lien tuc tren khoang (a; b) cha iem x0 va co ao hamtren (a; b)\{x0}

a) Neu f (x) oi dau t am sang dng khi x i qua x0 th f at cc tieu tai x0.

b) Neu f (x) oi dau t dng sang am khi x i qua x0 th f at cc ai tai x0.nh l 2: Gia s ham so f co ao ham tren khoang (a; b) cha iem x0, f (x0) = 0 va coao ham cap hai khac 0 tai iem x0.

a) Neu f (x0) < 0 th f at cc ai tai x0.

b) Neu f (x0) > 0 th f at cc tieu tai x0.

VAN E 1: Tm cc tr cua ham so

Qui tac 1: Dung nh l 1.

Tm f(x).

Tm cac iem xi (i = 1, 2, ) ma tai o ao ham bang 0 hoac khong co ao ham.

Xet dau f(x). Neu f(x) oi dau khi x i qua xi th ham so at cc tr tai xi.

Qui tac 2: Dung nh l 2.

Tnh f(x).

Giai phng trnh f(x) = 0 tm cac nghiem xi (i = 1, 2, ).

Tnh f(x) va f(xi) (i = 1, 2, ).Neu f(xi) < 0 th ham so at cc ai tai xi.

Neu f(xi) > 0 th ham so at cc tieu tai xi.

II. CC TR CUA HAM SO


8/51


Trang 7

Bai 1. Tm cc tr cua cac ham so sau:a) 2 33 2y x x= - b) 3 22 2 1y x x x= - + - c) 3 21 4 15

3y x x x= - + -

d)4

2 32

xy x= - + e) 4 24 5y x x= - + f)

42 3

2 2x

y x= - + +

g)2 3 6

2x x

yx

- + +=

+h)

23 4 51

x xy

x

+ +=

+i)

2 2 153

x xy

x

- -=

-

Bai 2. Tm cc tr cua cac ham so sau:a) 3 4( 2) ( 1)y x x= - + b)

2

24 2 1

2 3

x xy

x x

+ -=

+ -c)

2

23 4 4

1

x xy

x x

+ +=

+ +

d) 2 4y x x= - e) 2 2 5y x x= - + f) 22y x x x= + -

Bai 3. Tm cc tr cua cac ham so sau:a) 3 2 1y x= + b)

3 2

2 1xy

x=

+c) 4x xy e e-= +

d) 2 5 5 2 lny x x x= - + + e) 24siny x x= - f) 2ln(1 )y x x= - +

VAN E 2: Tm ieu kien e ham so co cc tr1.Neu ham so y = f(x) at cc tr tai iem x0 th f(x0) = 0 hoac tai x0 khong co ao ham.

2. e ham so y = f(x) at cc tr tai iem x0 th f(x) oi dau khi x i qua x0.Chu y:

Ham so bac ba 3 2y ax bx cx d = + + + co cc tr Phng trnh y= 0 co hai nghiem

phan biet.Khi o neu x0 la iem cc tr th ta co the tnh gia tr cc tr y(x0) bang hai cach:

+ 3 20 0 0 0( )y x ax bx cx d = + + +

+ 0 0( )y x Ax B= + , trong o Ax + B la phan d trong phep chia y cho y.

Ham so

2

' '

ax bx c

y a x b

+ +=

+ =

( )

( )

P x

Q x (aa 0) co cc tr Phng trnh y = 0 co hai

nghiem phan biet khac''

b

a- .

Khi o neu x0 la iem cc tr th ta co the tnh gia tr cc tr y(x0) bang hai cach:

00

0

( )( )

( )P x

y xQ x

= hoac 000

'( )( )

'( )P x

y xQ x

=

Khi s dung ieu kien can e xet ham so co cc tr can phai kiem tra lai e loai bonghiem ngoai lai. Khi giai cac bai tap loai nay thng ta con s dung cac kien thc khac na, nhat la

nh l Viet.


9/51


Trang 8

Bai 1. Chng minh rang cac ham so sau luon co cc ai, cc tieu:a) 3 2 2 33 3( 1)y x mx m x m= - + - - b) 3 22 3(2 1) 6 ( 1) 1y x m x m m x= - + + + +

c)2 2 4( 1) 1x m m x m

yx m

+ - - +=

-d)

2 21

x mx my

x m

+ - +=

- +

Bai 2.

Tm m e ham so:a) 3 2( 2) 3 5y m x x mx= + + + - co cc ai, cc tieu.

b) 3 2 23( 1) (2 3 2) ( 1)y x m x m m x m m= - - + - + - - co cc ai, cc tieu.

c) 3 2 23 ( 1) 2y x mx m x= - + - + at cc ai tai x = 2.

d) 4 22( 2) 5y mx m x m= - + - + - co mot cc ai 1 .2

x =

e)2 2 2x mx

yx m

- +=

-at cc tieu khi x = 2.

f)2 2

( 1) 4 21x m x m my

x- + - + -=

- co cc ai, cc tieu.

g)2

1x x m

yx

- +=

-co mot gia tr cc ai bang 0.

Bai 3. Tm m e cac ham so sau khong co cc tr:a) 3 23 3 3 4y x x mx m= - + + + b) 3 23 ( 1) 1y mx mx m x= + - - -

c)2 5

3x mx

yx

- + +=

-d)

2 2( 1) 4 21

x m x m my

x

- + - + -=

-

Bai 4. Tm a, b, c, d e ham so:a) 3 2y ax bx cx d = + + + at cc tieu bang 0 tai x = 0 va at cc ai bang

427

tai x =13

b) 4 2y ax bx c= + + co o th i qua goc toa o O va at cc tr bang 9 tai x = 3 .

c)2

1x bx c

yx

+ +=

-at cc tr bang 6 tai x = 1.

d)2ax bx ab

ybx a

+ +=

+at cc tr tai x = 0 va x = 4.

e)

2

2 2 1ax x by x

+ += + at cc ai bang 5 tai x = 1.

Bai 5. Tm m e ham so :a) 3 2 2 22( 1) ( 4 1) 2( 1)y x m x m m x m= + - + - + - + at cc tr tai hai iem x1, x2 sao

cho: 1 21 2

1 1 1( )

2x x

x x+ = + .

b) 3 21

13

y x mx mx= - + - at cc tr tai hai iem x1, x2 sao cho: 1 2 8x x- .

c) 3 21 1

( 1) 3( 2)3 3y mx m x m x= - - + - + at cc tr tai hai iem x1, x2 sao cho:

1 22 1x x+ = .


10/51


Trang 9

Bai 6. Tm m e ham so :a)

2 21

x mx my

x m

+ - +=

- +co cc ai, cc tieu va cac gia tr cc ai, cc tieu cung dau.

b)2 2( 1) 4 2

1x m x m m

yx

- + - + -=

-co cc ai, cc tieu va tch cac gia tr cc ai, cc

tieu at gia tr nho nhat.

c)2 3

4x x m

yx

- + +=

-co gia tr cc ai M va gia tr cc tieu m thoa 4M m- = .

d)22 3 2

2x x m

yx

+ + -=

+co 12C CT y y- < .

Bai 7. Tm m e o th ham so :a) 3 2 4y x mx= - + - co hai iem cc tr la A, B va

22 900

729m

AB = .

b) 4 2 4y x mx x m= - + + co 3 iem cc tr la A, B, C va tam giac ABC nhan goc toa oO lam trong tam.

c)2 2x mx m

yx m

+ + -=

-co hai iem cc tr nam hai pha oi vi truc tung. Chng minh

hai iem cc tr luon luon nam cung mot pha oi vi truc hoanh.

d)2

1x mx

yx

+=

-co khoang cach gia hai iem cc tr bang 10.

e)2 2 5

1x mx

yx

- + +=

-co hai iem cc ai va cc tieu nam ve hai pha oi vi ng

thang y = 2x.

f)2 2 3x x m

yx m

+ + +=

-co hai iem cc tr va khoang cach gia chung nho nhat.

Bai 8. Tm m e o th ham so :a) 3 22 12 13y x mx x= + - - co hai iem cc tr cach eu truc tung.

b) 3 2 33 4y x mx m= - + co cac iem cc ai, cc tieu oi xng nhau qua ng phangiac th nhat.c) 3 2 33 4y x mx m= - + co cac iem cc ai, cc tieu ve mot pha oi vi ngthang (d): 3 2 8 0x y- + = .

d)2 2(2 1) 1

1x m x m

yx

+ + + +=

+co hai iem cc tr nam hai pha oi vi ng thang

(d): 2 3 1 0x y- - = .

Bai 9. Tm m e o th ham so :a)

2 ( 1) 2 1x m x my

x m

- + + -=

-co hai iem cc tr trong goc phan t th nhat cua mat

phang toa o.

b)2 2 22 (4 1) 32 2

2mx m x m m

yx m

+ + + +=

+co mot iem cc tr nam trong goc phan t th

hai va iem kia nam trong goc phan t th t cua mat phang toa o.


11/51


Trang 10

c)2 2 2( 1) 4mx m x m m

yx m

- + + +=

-co mot iem cc tr nam trong goc phan t th nhat

va iem kia nam trong goc phan t th ba cua mat phang toa o.

d)2 2(2 1) 1

1x m x m

yx

+ + + +=

+co hai iem cc tr nam hai pha cua truc hoanh (tung).

VAN E 3: ng thang i qua hai iem cc tr

1) Ham so bac ba 3 2( )y f x ax bx cx d = = + + + .Chia f(x) cho f(x) ta c: f(x) = Q(x).f(x) + Ax + B.Khi o, gia s (x1; y1), (x2; y2) la cac iem cc tr th:

1 1 1

2 2 2

( )( )

y f x Ax B

y f x Ax B

= = +

= = +

Cac iem (x1; y1), (x2; y2) nam tren ng thang y = Ax + B.

2) Ham so phan thc2

( )( ) ( )P x ax bx cy f xQ x dx e

+ += = =+

.

Gia s (x0; y0) la iem cc tr th 000

'( )'( )

P xy

Q x= .

Gia s ham so co cc ai va cc tieu th phng trnh ng thang i qua hai iem

cc tr ay la:'( ) 2'( )

P x ax by

Q x d

+= = .

Bai 1. Viet phng trnh ng thang i qua hai iem cc tr cua o th ham so :a) 3 22 1y x x x= - - + b) 2 33 2y x x= - c) 3 23 6 8y x x x= - - +

d)22 1

3x x

yx

- +=

+e

2 12

x xy

x

- -=

-

Bai 2. Khi ham so co cc ai, cc tieu, viet phng trnh ng thang i qua hai iem cctr cua o th ham so:

a) 3 2 2 33 3( 1)y x mx m x m= - + - - b)2 6x mx

yx m

+ -=

-

c) 3 2 23( 1) (2 3 2) ( 1)y x m x m m x m m= - - + - + - - d)

2 21

x mx my x m

+ - += - +

Bai 3. Tm m e ham so:a) 3 22 3( 1) 6( 2) 1y x m x m x= + - + - - co ng thang i qua hai iem cc tr song songvi ng thang y = 4x + 1.b) 3 22 3( 1) 6 (1 2 )y x m x m m x= + - + - co cac iem cc ai, cc tieu cua o th nam trenng thang y = 4x.c) 3 2 7 3y x mx x= + + + co ng thang i qua cac iem cc ai, cc tieu vuong gocvi ng thang y = 3x 7.d) 3 2 23y x x m x m= - + + co cac iem cc ai va cc tieu oi xng nhau qua ng

thang (D):1 52 2

y x= - .


12/51


Trang 11

1. nh ngha:Gia s ham so f xac nh tren mien D (D R).

a)0 0

( ) ,max ( ) : ( )Df x M x D

M f xx D f x M

" = $ =

b)0 0

( ) ,min ( ) : ( )Df x m x D

m f xx D f x m

" = $ =

2. Tnh chat:a) Neu ham so f ong bien tren [a; b] th

[ ; ] [ ; ]

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

f x f b f x f a= = .

b) Neu ham so f nghch bien tren [a; b] th[ ; ] [ ; ]max ( ) ( ), min ( ) ( )

a b a bf x f a f x f b= = .

VAN E 1: Tm GTLN, GTNN cua ham so bang cach lap bang bien thienCach 1: Thng dung khi tm GTLN, GTNN cua ham so tren mot khoang.

Tnh f(x).Xet dau f(x) va lap bang bien thien.Da vao bang bien thien e ket luan.

Cach 2: Thng dung khi tm GTLN, GTNN cua ham solien tuc tren mot oan [a; b].Tnh f(x).Giai phng trnh f(x) = 0 tm c cac nghiem x1, x2, , xn tren [a; b] (neu co).Tnh f(a), f(b), f(x1), f(x2), , f(xn).So sanh cac gia tr va tnh va ket luan.

{ }1 2[ ; ]max ( ) max ( ), ( ), ( ), ( ),..., ( )n

a bM f x f a f b f x f x f x= =

{ }1 2[ ; ]min ( ) min ( ), ( ), ( ), ( ),..., ( )na b

m f x f a f b f x f x f x= =

Bai 1. Tm GTLN, GTNN cua cac ham so sau:a) 2 4 3y x x= + + b) 3 44 3y x x= - c) 4 22 2y x x= + -

d) 2 2y x x= + - e)2

1

2 2

xy

x x

-=

- +f)

2

22 4 5

1

x xy

x

+ +=

+

g) 21

( 0)y x xx

= + > h)2

21

1

x xy

x x

- +=

+ +i)

4 2

31 ( 0)x xy x

x x

+ += >

+

Bai 2. Tm GTLN, GTNN cua cac ham so sau:a) 3 22 3 12 1y x x x= + - + tren [1; 5] b) 33y x x= - tren [2; 3]

c) 4 22 3y x x= - + tren [3; 2] d) 4 22 5y x x= - + tren [2; 2]

e) 3 13

xy

x

-=

-tren [0; 2] f) 1

1x

yx

-=

+tren [0; 4]

III. GIA TR LN NHATVA GIA TR NHO NHAT CUA HAM SO


13/51


Trang 12

g)24 7 7

2x x

yx

+ +=

+tren [0; 2] h)

2

21

1

x xy

x x

- +=

+ -tren [0; 1]

i) 2100y x= - tren [6; 8] k) 2 4y x x= + + - Bai 3. Tm GTLN, GTNN cua cac ham so sau:

a)2sin 1sin 2

xy x

-= + b) 2

1

cos cos 1y x x= + + c) 22sin cos 1y x x= - +

d) cos2 2sin 1y x x= - - e) 3 3sin cosy x x= + f)2

4 21

1

xy

x x

-=

- +

g) 2 24 2 5 2 3y x x x x= - + + - + h) 2 24 4 3y x x x x= - + + - +

VAN E 2: Tm GTLN, GTNN cua ham so bang cach dung bat ang thcCach nay da trc tiep vao nh ngha GTLN, GTNN cua ham so.

Chng minh mot bat ang thc. Tm mot iem thuoc D sao cho ng vi gia tr ay, bat ang thc va tm c trthanh ang thc.

Bai 1. Gia s { }( ; ; ) / 0, 0, 0, 1D x y z x y z x y z= > > > + + = . Tm gia tr ln nhat cua bieuthc:

1 1 1x y z

Px y z

= + ++ + +

.

HD:1 1 1

3

1 1 1

P

x y z

= - + +

+ + +

S dung bat ang thc Cosi: [ ]1 1 1( 1) ( 1) ( 1) 9

1 1 1x y z

x y z

+ + + + = + +

+ + +

P 34

. Dau = xay ra x = y = z =13

. Vay3min4D

P = .

Bai 2. Cho D = 5( ; ) / 0, 0,4

x y x y x y

> > + =

. Tm gia tr nho nhat cua bieu thc:

4 14

Sx y

= + .

HD: ( ) 1 1 1 1 14 254

x x x x yx x x x y

+ + + + + + + +

4 14( ) 254

x yx y

+ +

S5. Dau = xay ra x = 1, y =14

. Vay minS = 5.

Bai 3. Cho D = { }( ; ) / 0, 0, 1x y x y x y> > + < . Tm gia tr nho nhat cua bieu thc:2 2 1

1 1x y

P x yx y x y

= + + + +- - +

.

HD:2 2 1(1 ) (1 ) 2

1 1

x yP x y

x y x y= + + + + + + -

- - +=

1 1 1 21 1x y x y

+ + -

- - +.

S dung bat ang thc Cosi: [ ]1 1 1

(1 ) (1 ) ( ) 91 1

x y x yx y x y

- + - + + + +

- - +


14/51


Trang 13

1 1 1 9

1 1 2x y x y+ +

- - +

P 52

. Dau = xay ra x = y =13

. Vay minP =52

.

Bai 4. Cho D = { }( ; ) / 0, 0, 4x y x y x y> > + . Tm gia tr nho nhat cua bieu thc:2 2

23 4 2

4x y

Px y

+ += + .

HD:2

1 12

4 8 8 2x y y x y

Px y

+= + + + + +

(1)

Theo bat ang thc Cosi:1 1

2 . 14 4x x

x x+ = (2)

32 2

1 1 33 . .

8 8 8 8 4y y y y

y y+ + = (3)

P 92

. Dau = xay ra x = y = 2. Vay minP =92

.

VAN E 3: Tm GTLN, GTNN cua ham so bang cach dung mien gia trXet bai toan tm GTLN, GTNN cua ham so f(x) tren mot mien D cho trc.

Goi y0 la mot gia tr tuy y cua f(x) tren D, th he phng trnh (an x) sau co nghiem:

0( ) (1)(2)

f x y

x D

=

Tuy theo dang cua he tren ma ta co cac ieu kien tng ng. Thong thng ieu kienay (sau khi bien oi) co dang: m y0M (3)V y0 la mot gia tr bat k cua f(x) nen t (3) ta suy ra c:

min ( ) ; max ( )D D

f x m f x M = =

Bai 1. Tm gia tr ln nhat, gia tr nho nhat cua cac ham so sau:a)

2

21

1

x xy

x x

+ +=

- +b)

2

22 7 23

2 10

x xy

x x

+ +=

+ +c)

2sin cos 1sin 2 cos 3

x xy

x x

+ +=

- +

d)2sin cos 32 cos sin 4

x xy x x

+ += - +

VAN E 4: S dung GTLN, GTNN cua ham so trong PT, HPT, BPTGia s f(x) la mot ham so lien tuc tren mien D va co min ( ) ; max ( )

D Df x m f x M = = . Khi o:

1) He phng trnh ( )f xx D

=

aco nghiem m aM.

2) He bat phng trnh( )f x

x D

aco nghiem Ma.

3) He bat phng trnh ( )f xx D

bco nghiem m b.


15/51


Trang 14

4) Bat phng trnh f(x) a ung vi moi x m a.5) Bat phng trnh f(x) bung vi moi x Mb.

Bai 1. Giai cac phng trnh sau:a) 4 42 4 2x x- + - = b) 3 5 6 2x x x+ = + c) 5 5 1(1 ) 16

x x+ - =

Bai 2. Tm m e cac phng trnh sau co nghiem:a) 22 1x x m+ + = b) 2 2 (2 )(2 )x x x x m- + + - - + =

c) 3 6 (3 )(6 )x x x x m+ + - - + - = d) 7 2 (7 )(2 )x x x x m- + + - - + =

Bai 3. Tm m e cac bat phng trnh sau nghiem ung vi moi x R:a) 22 1x x m+ + > b) 22 9m x x m+ < + c) 4 4 0mx x m- +

Bai 4. Cho bat phng trnh: 3 22 1 0x x x m- + - + < .a) Tm m e bat phng trnh co nghiem thuoc [0; 2].b) Tm m e bat phng trnh thoa moi x thuoc [0; 2].

Bai 5. Tm m e cac bat phng trnh sau:a) 3 1mx x m- - + co nghiem. b) ( 2) 1m x m x+ - + co nghiem x [0; 2].

c) 2 2( 1) 1m x x x x- + + + nghiem ung vi moi x [0; 1].


16/51


Trang 15

1. nh ngha:iem ( )0 0; ( )U x f x gl iem uon cua o th ham soy = f(x) neu ton tai mot khoang (a;

b) cha iem x0 sao cho tren mot trong hai khoang (a; x0) va (x0; b) tiep tuyen cua oth tai iem U nam pha tren o th con tren khoang kia tiep tuyen nam pha di o th

2. Tnh chat: Neu ham so y = f(x) co ao ham cap hai tren mot khoang cha iem x0, f(x0) = 0 vaf(x) oi dau khi x i qua x0 th ( )0 0; ( )U x f x la mot iem uon cua o th ham so.

o th cua ham so bac ba 3 2y ax bx cx d = + + + (a 0) luon co mot iem uon va ola tam oi xng cua o th.

Bai 1. Tm iem uon cua o th cac ham so sau:a) 3 26 3 2y x x x= - + + b) 3 23 9 9y x x x= - - + c) 4 26 3y x x= - +

d)4

22 34

xy x= - + e) 4 3 212 48 10y x x x= - + + f) 5 43 5 3 2y x x x= - + -

Bai 2. Tm m, n e o th cua ham so sau co iem uon c ch ra:a) 3 23 3 3 4y x x mx m= - + + + ; I(1; 2). b)

32 8( 1) ( 3)

3 3x

y m x m x= - + - + + - ; I(1; 3)

c) 3 2 1y mx nx= + + ; I(1; 4) d) 3 2 2y x mx nx= - + - ;2

; 33

I

-

e)3

23 2x

y mxm

= - + - ; I(1; 0) f) 3 23 4y mx mx= + + ; I(1; 2)

Bai 3. Tm m e o th cua cac ham so sau co 3 iem uon:a)

54 34 (4 3) 5 1

5 3x

y x m x x= - + + + - b)2

21

1

x mxy

x

+ -=

+

Bai 4. Chng minh o th cua cac ham so sau co 3 iem uon thang hang:a)

22 1

1

xy

x x

+=

+ +b)

21

1

xy

x

+=

+c)

2

22 3

1

x xy

x

-=

+

d)2

2 11

xyx

+=+

e)2 1xy

x=

+f) 2

22 5

1x xyx x

+ +=- +

g)2

22 3

3 3

x xy

x x

-=

- +h)

2

23

1

x xy

x

+=

+i)

3

2 4 5

xy

x x=

- +

Bai 5. Tm m, n e o th cua cac ham so:a) 4 3 22 6 2 1y x x x mx m= - - + + - co hai iem uon thang hang vi iem A(1; 2).

b)3

2 23 3

xy x mx= - - + + co iem uon tren ng thang 2y x= + .

c) 4 214y x mx n= - + + co iem uon tren Ox.

IV. IEM UON CUA O TH


17/51


Trang 16

1. nh ngha: ng thang 0x x= gl ng tiem can ng cua o th ham so ( )y f x= neu t

nhat mot trong cac ieu kien sau c thoa man:

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 thang 0y y= gl ng tiem can ngang cua o th ham so ( )y f x= neu t

nhat mot trong cac ieu kien sau c thoa man:

0lim ( )x

f x y+

= ; 0lim ( )x

f x y-

=

ng thang , 0y ax b a= + gl ng tiem can xien cua o th ham so ( )y f x= neu t nhat mot trong cac ieu kien sau c thoa man:

[ ]lim ( ) ( ) 0x

f x ax b+

- + = ; [ ]lim ( ) ( ) 0x

f x ax b-

- + =

2. Chu y:

a) Neu ( )( )( )

P xy f x

Q x= = la ham so phan thc hu ty.

Neu Q(x) = 0 co nghiem x0 th o th co tiem can ng 0x x= .

Neu bac(P(x)) bac(Q(x)) th o th co tiem can ngang. Neu bac(P(x)) = bac(Q(x)) + 1 th o th co tiem can xien.

b) e xac nh cac he so a, b trong phng trnh cua tiem can xien, ta co the ap dungcac cong thc sau:

[ ]( )lim ; lim ( )

x x

f xa b f x ax

x+ += = -

hoac [ ]( )

lim ; lim ( )x x

f xa b f x ax

x- -= = -

Bai 1. Tm cac tiem can cua o th cac ham so sau:a)

2 5

1

xy

x

-=

-

b)10 3

1 2

xy

x

+=

-

c)2 3

2

xy

x

+=

-

d)2 4 3

1x x

yx

- +=

+e)

2( 2)1

xy

x

-=

-f)

27 4 52 3

x xy

x

+ +=

-

Bai 2. Tm cac tiem can cua o th cac ham so sau:a)

2 4 5

xy

x x=

- +b)

22

9

xy

x

+=

-c)

2

24 5

1

x xy

x

+ +=

-

d)2

22 3 3

1

x xy

x x

+ +=

+ +e)

3

21

1

x xy

x

+ +=

+f)

4

34

1

x xy

x

- +=

-

Bai 3. Tm cac tiem can cua o th cac ham so sau:a) 2 4y x x= - b)

2

4 2

9

xy

x

+=

-c)

2

1

4 3y

x x=

- +

V. NG TIEM CAN CUA O TH


18/51


Trang 17

d) 11

xy x

x

-=

+e) 3 2 33y x x= - f)

2 3 22

x xy

x

- +=

-

Bai 4. Tm cac tiem can cua o th cac ham so sau:a) 2 1

2 1

x

xy

+=

-b) ln

2

x xe ey

--= c) 2ln( 5 6)y x x= - +

Bai 5. Tm m e o th cua cac ham so sau co ung hai tiem can ng:a)

23

2 2 1y

x mx m=

+ + -b)

2

22

3 2( 1) 4

xy

x m x

+=

+ + +c)

23

2

xy

x x m

+=

+ + -

Bai 6. Tm m e o th cua cac ham so sau co tiem can xien:a)

2 (3 2) 2 15

x m x my

x

+ + + -=

+b)

2 (2 1) 32

mx m x my

x

+ + + +=

+

Bai 7. Tnh dien tch cua tam giac tao bi tiem can xien cua o th cac ham so sau chantren hai truc toa o:

a)2

3 11x xy

x+ +=-

b)2

3 42x xyx

- + -=+

c)2

73x xy

x+ -=-

Bai 8. Tm m e tiem can xien cua o th cac ham so sau tao vi cac truc toa o mot tamgiac co dien tch S a ch ra:

a)2 1

1x mx

yx

+ -=

-; S = 8 b)

2 (2 1) 2 31

x m x my

x

+ - - +=

+; S = 8

c)22 2(2 1) 4 5

1x m x m

yx

+ + + -=

+; S = 16 d)

22 21

x mxy

x

+ -=

-; S = 4

Bai 9. Chng minh rang tch cac khoang cach t mot iem bat k tren o th cua cac hamso en hai tiem can bang mot hang so:a)

2 11

x xy

x

- +=

-b)

22 5 43

x xy

x

+ -=

+c)

2 73

x xy

x

+ -=

-


19/51


Trang 18

1. Cac bc khao sat s bien thien va ve o th cua ham so Tm tap xac nh cua ham so. Xet s bien thien cua ham so:

+ Tnh y.+ Tm cac iem tai o ao ham y bang 0 hoac khong xac nh.+ Tm cac gii han tai vo cc, gii han vo cc va tm tiem can (neu co).+ Lap bang bien thien ghi ro dau cua ao ham, chieu bien thien, cc tr cua ham so.

Ve o th cua ham so:+ Tm iem uon cua o th (oi vi ham so bac ba va ham so trung phng).

Tnh y. Tm cac iem tai o y = 0 va xet dau y.+ Ve cac ng tiem can (neu co) cua o th.+ Xac nh mot so iem ac biet cua o th nh giao iem cua o th vi cac truc

toa o (trong trng hp o th khong cat cac truc toa o hoac viec tm toa o giaoiem phc tap th co the bo qua). Co the tm them mot so iem thuoc o th e co theve chnh xac hn.

+ Nhan xet ve o th: Ch ra truc oi xng, tam oi xng (neu co) cua o th.

2. Ham so bac ba y 3 2 ( 0)y ax bx cx d a= + + + :

Tap xac nh R = R. o th luon co mot iem uon va nhan iem uon lam tam oi xng. Cac dang o th:

a > 0 a < 0y = 0 co 2 nghiem phan biet

D = b2 3ac > 0

y = 0 co nghiem kepD = b2 3ac = 0

y = 0 vo nghiemD = b2 3ac < 0

y

x0

I

y

x0

I

y

x0 I

y

x0

I

y

x0

I

VI. KHAO SAT S BIEN THIENVA VE O TH CUA HAM SO

y

x0

I


20/51


Trang 19

3. Ham so trung phng 4 2 ( 0)y ax bx c a= + + :

Tap xac nh D = R. o th luon nhan truc tung lam truc oi xng. Cac dang o th:

4. Ham so nhat bien ( 0, 0)ax by c ad bccx d

+= -

+:

Tap xac nh D = \ dRc

-

.

o th co mot tiem can ng la dxc

= - va mot tiem can ngang la ayc

= . Giao iem

cua hai tiem can la tam oi xng cua o th ham so. Cac dang o th:

5. Ham so hu ty2

( . ' 0, )' '

ax bx cy a a t khong chia het cho mau

a x b

+ +=

+:

Tap xac nh D = '\'

bR

a

-

.

o th co mot tiem can ng la ''

bx

a= - va mot tiem can xien. Giao iem cua hai

tiem can la tam oi xng cua o th ham so.

a > 0 a < 0

y = 0 co 3 nghiemphan biet ab < 0

y = 0 ch co1 nghiem ab > 0

y

x0

y

x0

y

x0

y

x0

0

ad bc > 0

x

y

0

ad bc < 0

x

y


21/51


Trang 20

Cac dang o th:

a.a > 0 a.a < 0

y = 0 co 2 nghiem phan biet

y = vo nghiem

Bai 1. Khao sat s bien thien va ve o th cua cac ham so: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 2y x x x= - - - +

Bai 2. Khao sat s bien thien va ve o th cua cac ham so:a) 4 22 1y x x= - - b) 4 24 1y x x= - + c)

42 53

2 2x

y x= - +

d) 2 2( 1) ( 1)y x x= - + e) 4 22 2y x x= - + + f) 4 22 4 8y x x= - + + Bai 3. Khao sat s bien thien va ve o th cua cac ham so:

a) 12

xy

x

+=

+b) 2 1

1x

yx

+=

-c) 3

4x

yx

-=

-

d)1 21 2

x

y x

-

= + e)3 1

3x

y x

-

= - f)2

2 1x

y x

-

= + Bai 4. Khao sat s bien thien va ve o th cua cac ham so:

a)2 1

1x x

yx

+ +=

+b)

2 21

x xy

x

+ +=

-c)

2 21

x xy

x

+ -=

+

d) 111

y xx

= - + +-

e)2

1x

yx

=-

f)2 2

1x x

yx

-=

+

Bai 5. Ve o th cua cac ham so:a)

33 2y x x= - + b) 3 23 2y x x= - + - c) 4 22 3y x x= - -

d) 11

xy

x

+=

-e)

2 21

x xy

x

- +=

-f)

2 3 32

x xy

x

+ +=

+

0 x

y

0 x

y

y

0 x0 x

y


22/51


Trang 21

1. S TNG GIAO CUA CAC O TH1. Cho hai o th (C1): y = f(x) va (C2): y = g(x). e tm hoanh o giao iem cua (C1) va

(C2) ta giai phng trnh: f(x) = g(x) (*) (goi la phng trnh hoanh o giao iem).So nghiem cua phng trnh (*) bang so giao iem cua hai o th.

2. o th ham so bac ba 3 2 ( 0)y ax bx cx d a= + + + cat truc hoanh tai 3 iem phan biet

Phng trnh 3 2 0ax bx cx d + + + = co 3 nghiem phan biet.

Ham so 3 2y ax bx cx d = + + + co cc ai, cc tieu va . 0C CT y y < .

Bai 1. Tm toa o giao iem cua cac o th cua cac ham so sau:

a)

2 33

2 21

2 2

xy x

xy

= - + -

= +

b)2

2 41

2 4

xy

xy x x

-=

- = - + +

c)34 3

2y x xy x

= -

= - +

d)4 2

21

4 5y x x

y x

= - +

= -e)

3 2

25 10 5

1y x x x

y x x

= - + -

= - +f)

2

13 1

xy

xy x

= - = - +

Bai 2. Bien luan theo m so giao iem cua cac o th cua cac ham so sau:

a)3 3 2( 2)

y x xy m x

= - +

= -b)

3 2

23 21 132 12

x xy x

y m x

= + -

= + +

c)3 3

3( 3)

xy x

y m x

= - + = -

d)2 1

22

xy

xy x m

+ = + = +

e)11

2

xy

xy x m

+ = - = - +

f)2 6 3

2x x

yx

y x m

- + = + = -

g)13

1

3

y xx

y mx

= - + + - = +

h)2 3 3

2

4 1

x xy

x

y mx m

- + = - = - -

i)3

22 1

( 1)y x x

y m x

= + +

= -

Bai 3. Tm m e o th cac ham so:a)

2( 2) 1; 12

xy y mx

x

+ -= = +

+cat nhau tai hai iem phan biet.

b)22 3

; 21

x x my y x m

x

- += = +

-cat nhau tai hai iem phan biet.

c)2

; 21

mx x my y mx

x

+ += = +

-cat nhau tai hai iem co hoanh o trai dau.

d)

2 4 5

; 22

x x

y y mxx

+ +

= = ++ cat nhau tai hai iem co hoanh o trai dau.

e)2( 2) ; 3

1x

y y mxx

-= = +

-cat nhau tai hai iem thuoc hai nhanh khac nhau.

VII. MOT SO BAI TOAN LIEN QUANEN KHAO SAT HAM SO


23/51


Trang 22

f)2

1mx x m

yx

+ +=

-cat truc hoanh tai hai iem phan biet co hoanh o dng.

Bai 4. Tm m e o th cac ham so:a) 3 23 2 ; 2y x x mx m y x= + + + = - + cat nhau tai ba iem phan biet.

b) 3 23 (1 2 ) 1y mx mx m x= + - - - cat truc hoanh tai ba iem phan biet.

c) 2 2( 1)( 3)y x x mx m= - - + - cat truc hoanh tai ba iem phan biet.

d) 3 2 22 2 2 1; 2 2y x x x m y x x= + - + - = - + cat nhau tai ba iem phan biet.

e) 3 2 2 22 3 ; 2 1y x x m x m y x= + - + = + cat nhau tai ba iem phan biet.Bai 5. Tm m e o th cac ham so:

a) 4 22 1;y x x y m= - - = cat nhau tai bon iem phan biet.

b) 4 2 3( 1)y x m m x m= - + + cat truc hoanh tai bon iem phan biet.

c) 4 2 2(2 3) 3y x m x m m= - - + - cat truc hoanh tai bon iem phan biet.

Bai 6. Tm m e o th cua cac ham so:a) 3 1; 2

4x

y y x mx

+= = +

-cat nhau tai hai iem phan biet A, B. Khi o tm m e oan

AB ngan nhat.

b)4 1

;2x

y y x mx

-= = - +

-cat nhau tai hai iem phan biet A, B. Khi o tm m e oan

AB ngan nhat.

c)2 2 4

; 2 22

x xy y mx m

x

- += = + -

-cat nhau tai hai iem phan biet A, B. Khi o tnh

AB theo m.Bai 7. Tm m e o th cua cac ham so:

a) 3 23 6 8y x mx mx= - + - cat truc hoanh tai ba iem co hoanh o lap thanh mot capso cong.b) 3 23 9 1; 4y x x x y x m= - - + = + cat nhau tai ba iem A, B, C vi B la trung iemcua oan AC.c) 4 2 2(2 4)y x m x m= - + + cat truc hoanh tai bon iem co hoanh o lap thanh mot capso cong.

d)3 2

( 1) ( 1) 2 1y x m x m x m= - + - - + - cat truc hoanh tai ba iem co hoanh o lapthanh mot cap so nhan.e) 3 23 (2 2) 9 192y x m x mx= + + + + cat truc hoanh tai ba iem co hoanh o lap thanhmot cap so nhan.


24/51


Trang 23

2. BIEN LUAN SO NGHIEM CUA PHNG TRNH BANG O TH

C s cua phng phap: Xet phng trnh: f(x) = g(x) (1)So nghiem cua phng trnh (1) = So giao iem cua (C1): y = f(x) va (C2): y = g(x)

Nghiem cua phng trnh (1) la hoanh o giao iem cua (C1): y = f(x) va (C2): y = g(x)

e bien luan so nghiem cua phng trnh F(x, m) = 0 (*) bang o th ta bien oi (*) vemot trong cac dang sau:

Dang 1: F(x, m) = 0 f(x) = m (1)Khi o (1) co the xem la phng trnh hoanh ogiao iem cua hai ng:

(C): y = f(x)d: y = m

d la ng thang cung phng vi truc hoanh. Da vao o th (C) ta bien luan so giao iemcua (C) va d. T o suy ra so nghiem cua (1)

Dang 2: F(x, m) = 0 f(x) = g(m) (2)Thc hien tng t nh tren, co the at g(m) = k.Bien luan theo k, sau o bien luan theo m.

Dang 3: F(x, m) = 0 f(x) = kx + m (3)(k: khong oi)

Khi o (3) co the xem la phng trnh hoanh ogiao iem cua hai ng:

(C): y = f(x)

d: y = kx + m V d co he so goc k khong oi nen d cung phngvi ng thang y = kx va cat truc tung tai iem A(0; m). Viet phng trnh cac tiep tuyen d1, d2, cua (C)co he so goc k. Da vao cac tung o goc b, b1, b2, cua d, d1, d2, e bien luan.

Dang 4: F(x, m) = 0 f(x) = m(x x0) + y0 (4)Khi o (4) co the xem la phng trnhhoanh o giao iem cua hai ng:

(C): y = f(x)d: y = m(x x0) + y0

d quay quanh iem co nh M0(x0; y0). Viet phng trnh cac tiep tuyen d1, d2, cua (C) i qua M0. Cho d quay quanh iem M0 e bien luan.

Chu y:

Neu F(x, m) = 0 co nghiem thoa ieu kien: ax b th ta ch ve o th (C): y = f(x)vi ax b.Neu co at an so phu th ta tm ieu kien cua an so phu, sau o bien luan theo m.

y

x

m A(C)(d) : y = m

yC

yCTxA

y

xA

y = kx

mC)

M1

M2

b1

b2

d1

dd2

O

y

x0

d3

d1 0

0

(C)

M1

M2

d2m =

m = +

m > 0

m = 0

m < 0

dI

IV()

(+)M

x


25/51


Trang 24

VAN E 1: Bien luan so nghiem cua phng trnh bang o th e bien luan so nghiem cua phng trnh F(x, m) = 0 (*) ta bien oi (*) ve mot trong cac

dang nh tren, trong o lu y y = f(x) la ham so a khao sat va ve o th.

Bai 1. Khao sat s bien thien va ve o th (C) cua ham so. Dung o th (C) bien luan theom so nghiem cua phng trnh:a) 3 33 1; 3 1 0y x x x x m= - + - + - = b) 3 33 1; 3 1 0y x x x x m= - + - - + + =

c) 3 3 23 1; 3 2 2 0y x x x x m m= - + - - - - = d) 3 33 1; 3 4 0y x x x x m= - + - - + + =

e)4

2 4 22 2; 4 4 2 02

xy x x x m= - + + - - + = f) 4 2 4 22 2; 2 2 0y x x x x m= - + - - + =

Bai 2. Khao sat s bien thien va ve o th (C) cua ham so. Dung o th (C) bien luan theom so nghiem cua phng trnh:

a)2

25 7 ; ( 5) 3 7 03

x xy x m x m

x

- += - + + + =

-

b)2

22 4 2 ; 2 2( 2) 3 2 02 3

x xy x m x m

x

- += - + - + =

+

c)2

21; ( 1) 2 1 0xy m x xx

+= - + - =

d)2

22 4 ; 2( 1) 4( 1) 02 4

x xy x m x m

x

- += - + + + =

-

Bai 3. Khao sat s bien thien va ve o th (C) cua ham so. Dung o th (C) bien luan theom so nghiem cua phng trnh:

a)2

22 ; 2sin 2 cos 2 0 (0 )2 1xy m m

x= + - - =

-a a a p

b)22 3

; cos2 ( 3) cos 2 1 0 (0 )2

x xy m m

x

-= - + + + =

-a a a p

c)2

23 3 ; cos (3 )cos 3 2 0 (0 )2

x xy m m

x

+ += + - + - =

+a a a p

d) 3 2 3 23 6; cos 3cos 6 0y x x x x m= - + - + - = Bai 4. Khao sat s bien thien va ve o th (C) cua ham so. Dung o th (C) bien luan theo

m so nghiem cua phng trnh:

a)2 5 7 ; 2 (3 7)2 5

3t tx xy m m

x-- += + + = +

-

b)2 1; 2 ( 1)2 1

1t tx xy m m

x-+ -= + - = -

-

c)2

22 5 4 ; 2 (5 ) 4 01

t tx xy e m e mx

- += - + + + =

-

d)2

25 4 ; (5 ) 4 0t tx x

y e m ex

- += - + + =

Bai 5.

Khao sat s bien thien va ve o th (C) cua ham so. T o th (C) hay suy ra o th(T). Dung o th (T) bien luan theo m so nghiem cua phng trnh:

a)2 2 23 6 3 6 3 6( ) : ; ( ) : ; 2 0

1 1 1x x x x x x

C y T y mx x x

- + - + - += = - =

- - -


26/51


Trang 25

b)2 2 25 4 5 4 5 4( ) : ; ( ) : ; 2 0x x x x x xC y T y m

x x x

- + - + - += = - + =

c) 3 2 3 2 3 2( ) : 3 6; ( ) : 3 6 ; 3 6 3 0C y x x T y x x x x m= - + = - + - + - + =

d)3 33 2 2 2( ) : 2 9 12 4; ( ) : 2 9 12 4; 2 9 12 0C y x x x T y x x x x x x m= - + - = - + - - + + =

e) 2 2 2 2( ) : ( 1) (2 ); ( ) : ( 1) 2 ;( 1) 2 ( 1) (2 )C y x x T y x x x x m m= + - = + - + - = + -

f)2 2

21 1( ) : ; ( ) : ; ( 1) 2 1 0x xC y T y m x xx x

+ += = - + - =

Bai 6. Cho ham so 2( )1

xy f x

x

+= =

-.

a) Khao sat s bien thien va ve o th (C) cua ham so.b) Viet phng trnh tiep tuyen cua (C) vuong goc vi ng thang 3 0x y- = .c) Dung o th (C), bien luan so nghiem cua phng trnh:

2

3 ( 2) 2 0x m x m- + + + =

Bai 7. Cho ham so 1( )

1x

y f xx

+= =

-.

a) Khao sat s bien thien va ve o th (C) cua ham so.b) Viet phng trnh tiep tuyen cua (C) vuong goc vi ng thang 2 0x y- = .c) Dung o th (C), bien luan theo m so nghiem cua phng trnh:

22 ( 1) 1 0x m x m- + + + =

Bai 8. Cho ham so 2( )1

xy f x

x= =

-.

a) Khao sat s bien thien va ve o th (C) cua ham so.b) Viet phng trnh tiep tuyen cua (C) i qua iem A(0; 1).c) Dung o th (C), bien luan theo m so nghiem cua phng trnh:

2(1 ) (1 ) 1 0m x m x- - - + =

VAN E 2: Bien luan so nghiem cua phng trnh bac ba bang o th

C s cua phng phap: Xet phng trnh bac ba: 3 2 0ax bx cx d + + + = (a 0) (1)

Goi (C) la o th cua ham so bac ba: 3 2( )y f x ax bx cx d = = + + + So nghiem cua (1) = So giao iem cua (C) vi truc hoanh

Dang 1:Bien luan so nghiem cua phng trnh bac 3Trng hp 1: (1) ch co 1 nghiem (C) va Ox co 1 iem chung

( .1 )

2 ( .1 ). 0C CT

f khong co cc tr h af co cc tr

h by y


27/51


Trang 26

Trng hp 2: (1) co ung 2 nghiem (C) tiep xuc vi Ox

2 ( .2). 0C CT

f co cc trh

y y =

Trng hp 3: (1) co 3 nghiem phan biet(C) cat Ox tai 3 iem phan biet

2 ( .3). 0C CT

f co cc trh

y y < 0

yCT

B

f(0)

x1xA xB xC

C

(C)

yC

y

Ao x2 x

a < 0

yCT

Bf(0)

x"0C

x1

(C)yC

y

Ao

x2x

(H.3)yC

x0 x'0B

(C)yC

y

Ax0o x1

Bx'0

(yCT = f(x0) = 0)

x

(H.2)

x1xA xB xCC

(C)

yC

y

Ao

x2x

a > 0

yCT

B

f(0)

xCx2x1

xA xBC

(C)

yC

y

Ao x

a < 0

yCT

B

f(0)


28/51


Trang 27

Bai 1. Tm m e cac phng trnh sau ch co 1 nghiem:a) 3 22 3( 1) 6 2 0x m x mx- + + - = b) 3 23 3(1 ) 1 3 0x x m x m- + - + + =

c) 3 22 3 6( 1) 3 12 0x mx m x m- + - - + = d) 3 26 3( 4) 4 8 0x x m x m- - - + - =

e) 3 22 3( 1) 6( 2) 2 0x m x m x m+ - + - + - = f) 3 3 2 0x mx m- + =

Bai 2.

Tm m e cac phng trnh sau ch co 2 nghiem:a) 3 2 2( 1) (2 3 2) 2 (2 1) 0x m x m m x m m- + - - + + - = b) 3 3 2 0x mx m- + =

c) 3 2(2 1) (3 1) ( 1) 0x m x m x m- + + + - + = d) 3 23 3(1 ) 1 3 0x x m x m- + - + + = Bai 3. Tm m e cac phng trnh sau co 3 nghiem phan biet:

a) 3 2 2 23 3( 1) ( 1) 0x mx m x m- + - - - = b) 3 26 3( 4) 4 8 0x x m x m- - - + - =

c) 3 22 3( 1) 6( 2) 2 0x m x m x m+ - + - + - = d) 31 03x x m- + =

Bai 4. Tm m e cac phng trnh sau co 3 nghiem dng phan biet:a) 3 2 2 23 3( 1) ( 1) 0x mx m x m- + - - - = b) 3 26 3( 4) 4 8 0x x m x m- - - + - =

c) 3 21 5 74 03 2 6x x x m- + + + = d) 3 2 (2 1) 2 0x mx m x m- + + - - =

Bai 5. Tm m e cac phng trnh sau co 3 nghiem am phan biet:a) 3 22 3( 1) 6( 2) 2 0x m x m x m+ - + - + - = b) 3 2 2 23 3( 1) ( 1) 0x mx m x m- + - - - =

c) 3 23 9 0x x x m+ - + = d) 3 2 18 2 0x x mx m- + - =


29/51


Trang 28

3. S TIEP XUC CUA HAI NG. TIEP TUYEN CUA NG CONG.

1. Y ngha hnh hoc cua ao ham: ao ham cua ham so y = f(x) tai iem x0 la he so goccua tiep tuyen vi o th (C) cua ham so tai iem ( )0 0 0; ( )M x f x .

Khi o phng trnh tiep tuyen cua (C) tai iem ( )0 0 0; ( )M x f x la:y y0 = f (x0).(x x0)

2. ieu kien can va u e hai ng (C1): y = f(x) va (C2): y = g(x) tiep xuc nhau la hephng trnh sau co nghiem:

( ) ( )'( ) '( )

f x g xf x g x

=

=(*)

Nghiem cua he (*) la hoanh o cua tiep iem cua hai ng o.3. Neu (C1): y = px + q va (C2): y = ax2 + bx + c th

(C1) va (C2) tiep xuc nhau phng trnh 2ax bx c px q+ + = + co nghiem kep.

VAN E 1: Lap phng trnh tiep tuyen cua ng cong (C): y = f(x) Bai toan 1: Viet phng trnh tiep tuyen D cua (C): y =f(x) tai iem ( )0 0 0;M x y :

Neu cho x0 th tm y0 = f(x0).Neu cho y0 th tm x0 la nghiem cua phng trnh f(x) = y0.

Tnh y= f(x0). Suy ra y(x0) = f(x0).Phng trnh tiep tuyen D la: y y0 = f(x0).(x x0)

Bai toan 2: Viet phng trnh tiep tuyen D cua (C): y =f(x), bietD co he so goc k cho trc.Cach 1: Tm toa o tiep iem.Goi M(x

0; y

0) la tiep iem. Tnh f(x

0).

D co he so goc kf(x0) = k (1)Giai phng trnh (1), tm c x0 va tnh y0 = f(x0). T o viet phng trnh cua D.

Cach 2: Dung ieu kien tiep xuc.Phng trnh ng thang D co dang: y = kx + m.D tiep xuc vi (C) khi va ch khi he phng trnh sau co nghiem:

( )'( )

f x kx mf x k

= + =

(*)

Giai he (*), tm c m. T o viet phng trnh cua D.

Chu y: He so goc k cua tiep tuyen D co the c cho gian tiep nh sau:

+ D tao vi chieu dng truc hoanh goc a th k = tana+ D song song vi ng thang d: y = ax + b th k = a

+ D vuong goc vi ng thang d: y = ax + b (a 0) th k =1a

-

+ D tao vi ng thang d: y = ax + b mot goc a th tan1k a

ka

-=

+a

Bai toan 3: Viet phng trnh tiep tuyen D cua (C): y = f(x), bietDi qua iem ( ; )A AA x y .

Cach 1: Tm toa o tiep iem.Goi M(x0; y0) la tiep iem. Khi o: y0 = f(x0), y0 = f(x0).

Phng trnh tiep tuyen D tai M: y y0 = f(x0).(x x0)D i qua ( ; )A AA x y nen: y A y0 = f(x0).(xA x0) (2)

Giai phng trnh (2), tm c x0. T o viet phng trnh cua D.


30/51


Trang 29

Cach 2: Dung ieu kien tiep xuc.Phng trnh ng thang D i qua ( ; )A AA x y va co he so goc k: y yA = k(x xA)

D tiep xuc vi (C) khi va ch khi he phng trnh sau co nghiem:

( ) ( )'( )

A Af x k x x y

f x k

= - +

=(*)

Giai he (*), tm c x (suy ra k). T o viet phng trnh tiep tuyen D.

Bai 6. Viet phng trnh tiep tuyen cua (C) tai iem c ch ra:a) (C): 3 23 7 1y x x x= - - + tai A(0; 1) b) (C): 4 22 1y x x= - + tai B(1; 0)

c) (C):3 42 3x

yx

+=

-tai C(1; 7) d) (C):

21

2 1y x

x= + -

-tai D(0; 3)

Bai 7. Viet phng trnh tiep tuyen cua (C) tai iem c ch ra:a) (C):

2 3 3

2

x xy

x

- +=

-tai iem A co xA = 4

b) (C):3( 2)

1x

yx

-=

-tai iem B co yB = 4

c) (C): 12

xy

x

+=

-tai cac giao iem cua (C) vi truc hoanh, truc tung.

d) (C): 22 2 1y x x= - + tai cac giao iem cua (C) vi truc hoanh, truc tung.

e) (C): 3 3 1y x x= - + tai iem uon cua (C).

f) (C): 4 21 924 4

y x x= - - tai cac giao iem cua (C) vi truc hoanh.

Bai 8. Viet phng trnh tiep tuyen cua (C) tai cac giao iem cua (C) vi ng c chra:a) (C): 3 22 3 9 4y x x x= - + - va d: 7 4y x= + .

b) (C): 3 22 3 9 4y x x x= - + - va (P): 2 8 3y x x= - + - .

c) (C): 3 22 3 9 4y x x x= - + - va (C): 3 24 6 7y x x x= - + - .Bai 9. Tnh dien tch tam giac chan hai truc toa o bi tiep tuyen cua o th (C) tai iem

c ch ra:

a) (C): 5 11

2 3

xy

x

+=

-

tai iem A co xA = 2 .

b) (C): 2 7 26y x x= - + tai iem B co xB = 2.Bai 10.Tm m e tiep tuyen cua o th (C) tai iem c ch ra chan hai truc toa o mot

tam giac co dien tch bang S cho trc:

a) (C): 21

x my

x

+=

-tai iem A co xA = 2 va S =

12

.

b) (C):32

x my

x

-=

+tai iem B co xB = 1 va S =

92

.

c) (C): 3 1 ( 1)y x m x= + - + tai iem C co xC = 0 va S = 8.

Bai 11.Viet phng trnh tiep tuyen D cua (C), biet D co he so goc k c ch ra:a) (C): 3 22 2 5y x x= - + ; k = 12 b) (C): 2 1

2x

yx

-=

-; k = 3


31/51


Trang 30

c) (C):2 3 4

1x x

yx

- +=

-; k = 1 d) (C): 2 4 3y x x= - + ; k = 2

Bai 12.Viet phng trnh tiep tuyen D cua (C), biet D song song vi ng thang d chotrc:

a) (C):3

22 3 13

xy x x= - + + ; d: y = 3x + 2 b) (C): 2 1

2

xy

x

-=

-; d: 3 2

4y x= - +

c) (C):2 2 34 6

x xy

x

- -=

+; d: 2 5 0x y+ - = d) (C): 4 2

1 33

2 2y x x= - + ; d: y = 4x + 1

Bai 13.Viet phng trnh tiep tuyen D cua (C), biet D vuong goc vi ng thang d chotrc:

a) (C):3

22 3 13

xy x x= - + + ; d: 2

8x

y = - + b) (C):2 1

2x

yx

-=

-; d: y x=

c) (C):2 3

1x

yx

+=

+; d: y = 3x d) (C):

2 12

x xy

x

+ -=

+; d: x 2

Bai 14.Viet phng trnh tiep tuyen D cua (C), biet D tao vi chieu dng truc Ox goc a:a) (C):

32 02 4; 60

3x

y x x= - + - =a b) (C):3

2 02 4; 753

xy x x= - + - =a

c) 03 2

( ) : ; 451

xC y

x

-= =

-a

Bai 15.Viet phng trnh tiep tuyen D cua (C), biet D tao vi ng thang d mot goc a:a) (C):

32 02 4; : 3 7; 45

3x

y x x d y x= - + - = + =a

b) (C):

3

2 012 4; : 3; 303 2xy x x d y x= - + - = - + =a

c) 04 3( ) : ; : 3 ; 451

xC y d y x

x

-= = =

-a

d) 03 7( ) : ; : ; 602 5x

C y d y xx

-= = - =

- +a

e)2

03( ) : ; : 1; 602

x xC y d y x

x

- += = - + =

-a

Bai 16.Tm m e tiep tuyen D cua (C) tai iem c ch ra vuong goc vi ng thang dcho trc:a) (C):

2 (2 1) 21

x m x my

x

+ + - +=

+tai iem A co xA = 0 va d la tiem can xien cua (C).

b) (C):22 1

3x mx

yx

+ -=

-; tai iem B co xB = 4 va d: x 12y + 1 = 0 .

Bai 17.Tm m e tiep tuyen D cua (C) tai iem c ch ra song song vi ng thang dcho trc:

a) (C):2(3 1) ( 0)m x m my m

x m

+ - +=

+tai iem A co yA = 0 va d: 10y x= - .

Bai 18.Viet phng trnh tiep tuyen D cua (C), biet Di qua iem c ch ra:a) (C): 3 3 2y x x= - + - ; A(2; 4) b) (C): 3 3 1y x x= - + ; B(1; 6)


32/51


Trang 31

c) (C): ( )222y x= - ; C(0; 4) d) (C): 4 21 33

2 2y x x= - + ; 30;

2D

e) (C):22

xy

x

+=

-; E(6; 5) f) (C):

3 41

xy

x

+=

-; F(2; 3)

g) (C):

2 3 3

2

x x

y x

- +=

- ; G(1; 0) h)

2 2

1

x x

y x

- +=

- ; H(2; 2)

VAN E 2: Tm ieu kien e hai ng tiep xuc 1. ieu kien can va u e hai ng (C1): y = f(x) va (C2): y = g(x) tiep xuc nhau la he

phng trnh sau co nghiem:

( ) ( )'( ) '( )

f x g xf x g x

=

=(*)

Nghiem cua he (*) la hoanh o cua tiep iem cua hai ng o.

2. Neu (C1): y = px + q va (C2): y = ax2 + bx + c th

(C1) va (C2) tiep xuc nhau phng trnh 2ax bx c px q+ + = + co nghiem kep.

Bai 1. Tm m e hai ng (C1), (C2) tiep xuc nhau:a) 3 21 2( ) : (3 ) 2; ( ) :C y x m x mx C truc hoanh= + + + +

b) 3 21 2( ) : 2 ( 1) ; ( ) :C y x x m x m C truc hoanh= - - - +

c) 31 2( ) : ( 1) 1; ( ) : 1C y x m x C y x= + + + = +

d) 3 21 2( ) : 2 2 1; ( ) :C y x x x C y x m= + + - = +

Bai 2. Tm m e hai ng (C1), (C2) tiep xuc nhau:a) 4 2 21 2( ) : 2 1; ( ) : 2C y x x C y mx m= + + = + b) 4 2 21 2( ) : 1; ( ) :C y x x C y x m= - + - = - +

c) 4 2 21 21 9( ) : 2 ; ( ) :4 4

C y x x C y x m= - + + = - +

d) 2 2 21 2( ) : ( 1) ( 1) ; ( ) : 2C y x x C y x m= + - = +

e)2

1 2(2 1)( ) : ; ( ) :

1m x m

C y C y xx

- -= =

-

f) 2 21 21( ) : ; ( ) :1x xC y C y x m

x- += = +-

VAN E 3: Lap phng trnh tiep tuyen chung cua hai o th(C1): y = f(x) va C2): y = g(x)

1. Goi D: y = ax + b la tiep tuyen chung cua (C1) va (C2).u la hoanh o tiep iem cua D va (C1), v la hoanh o tiep iem cua D va (C2).D tiep xuc vi (C=1) va (C2) khi va ch khi he sau co nghiem:

( ) (1)

'( ) (2)( ) (3)'( ) (4)

f u au b

f u ag v av bg v a

= + = = +

=


33/51


Trang 32

T (2) va (4) f(u) = g(v) u = h(v) (5)The a t (2) vao (1) b = j(u) (6)The (2), (5), (6) vao (3) v a u b. T o viet phng trnh cua D.

2. Neu (C1) va (C2) tiep xuc nhau tai iem co hoanh o x0 th mot tiep tuyen chung cua (C1)va (C2) cung la tiep tuyen cua (C1) (va (C2)) tai iem o.

Bai 1. Viet phng trnh tiep tuyen chung cua hai o th:a) 2 21 2( ) : 5 6; ( ) : 5 11C y x x C y x x= - + = - + -

b) 2 21 2( ) : 5 6; ( ) : 14C y x x C y x x= - + = - - -

c) 2 31 2( ) : 5 6; ( ) : 3 10C y x x C y x x= - + = + -

VAN E 4: Tm nhng iem tren o th (C): y = f(x) sao cho tai otiep tuyen cua (C) song song hoac vuong goc vi mot ng thang d cho trc

Goi M(x0; y0) (C). D la tiep tuyen cua (C) tai M. Tnh f(x0).V D // d nen f(x0) = kd (1)

hoac D^ d nen f(x0) =1

dk- (2)

Giai phng trnh (1) hoac (2) tm c x0. T o tm c M(x0; y0) (C).

Bai 1. Tm cac iem tren o th (C) ma tiep tuyen tai o vuong goc vi ng thang d chotrc:

a) (C):2 3 6

1

x xy

x

+ +=

+

; d:1

3y x=

b) (C):2 1

1x x

yx

+ +=

+; d la tiem can xien cua (C)

c) (C):2 1

1x x

yx

+ -=

-; d la ng thang i qua hai iem cc ai, cc tieu cua (C).

d) (C):2 1x x

yx

- += ; d: y = x

Bai 2. Tm cac iem tren o th (C) ma tiep tuyen tai o song song vi ng thang d chotrc:

a) (C): 3 2 10y x x x= + + + ; d: 2y x= b) (C): 2 1x xyx- += ; d: y = x

VAN E 5: Tm nhng iem tren ng thang d ma t o co the ve c1, 2, 3, tiep tuyen vi o th (C): y = f(x)

Gia s d: ax + by +c = 0. M(xM; yM) d.Phng trnh ng thang D qua M co he so goc k: y = k(x xM) + yMD tiep xuc vi (C) khi he sau co nghiem:

( ) ( ) (1)

'( ) (2)

M Mf x k x x y

f x k

= - +

=

The k t (2) vao (1) ta c: f(x) = (x xM).f(x) + yM (3)So tiep tuyen cua (C) ve t M = So nghiem x cua (3)


34/51


Trang 33

Bai 1. Tm cac iem tren o th (C) ma t o ve c ung mot tiep tuyen vi (C):a) 3 2( ) : 3 2C y x x= - + - b) 3( ) : 3 1C y x x= - +

Bai 2. Tm cac iem tren ng thang d ma t o ve c ung mot tiep tuyen vi (C):a)

1( ) :

1x

C yx

+=

-; d la truc tung b)

2 2( ) :

1x x

C yx

+ +=

-; d la truc hoanh

c)22( ) :

1x x

C yx

+=

+; d: y = 1 d)

2 3 3( ) :2

x xC y

x

+ +=

+; d: x = 1

e) 3( ) :1

xC y

x

+=

-; d: y = 2x + 1

Bai 3. Tm cac iem tren ng thang d ma t o ve c t nhat mot tiep tuyen vi (C):a)

2 6 9( ) :2

x xC y

x

- +=

- +; d la truc tung b)

2 3 3( ) :1

x xC y

x

+ +=

+; d la truc tung

c) 2 1( ) :2

xC y

x

+=

-; d: x = 3 d) 3 4( ) :

4 3x

C yx

+=

-; d: y = 2

Bai 4. Tm cac iem tren ng thang d ma t o ve c hai tiep tuyen vi (C):a)

2 2( ) :

2x x

C yx

+ -=

+; d la truc hoanh b)

2 1( ) :

1x x

C yx

- -=

+; d la truc tung

c)2 3 3( ) :

2x x

C yx

+ +=

+; d: y = 5

Bai 5. Tm cac iem tren ng thang d ma t o ve c ba tiep tuyen vi (C):a) 3 2( ) : 3 2C y x x= - + - ; d: y = 2 b) 3( ) : 3C y x x= - ; d: x = 2

c) 3( ) : 3 2C y x x= - + + ; d la truc hoanh d) 3( ) : 12 12C y x x= - + ; d: y = 4

e) 4 2( ) : 2C y x x= - - ; d la truc tung e) 4 2( ) : 2 1C y x x= - + - ; d la truc tungBai 6. T iem A co the ke c bao nhieu tiep tuyen vi (C):

a) 3 2( ) : 9 17 2C y x x x= - + + ; A(2; 5) b) 3 21 4 4( ) : 2 3 4; ;3 9 3

C y x x x A

= - + +

c) 3 2( ) : 2 3 5; (1; 4)C y x x A= + - - Bai 7. T mot iem bat k tren ng thang d co the ke c bao nhieu tiep tuyen vi (C):

a) 3 2( ) : 6 9 1C y x x x= - + - ; d: x = 2 b) 3( ) : 3C y x x= - ; d: x = 2

VAN E 6: Tm nhng iem ma t o co the ve c2 tiep tuyen vi o th (C): y = f(x) va 2 tiep tuyen o vuong goc vi nhau

Goi M(xM; yM).Phng trnh ng thang D qua M co he so goc k: y = k(x xM) + yMD tiep xuc vi (C) khi he sau co nghiem:

( ) ( ) (1)'( ) (2)

M Mf x k x x y

f x k

= - +

=

The k t (2) vao (1) ta c: f(x) = (x xM).f(x) + yM (3)Qua M ve c 2 tiep tuyen vi (C) (3) co 2 nghiem phan biet x1, x2.Hai tiep tuyen o vuong goc vi nhau f(x1).f(x2) = 1T o tm c M.


35/51


Trang 34

Chu y: Qua M ve c 2 tiep tuyen vi (C) sao cho 2 tiep iem nam ve hai pha vi truc

hoanh th1 2

(3) 2( ). ( ) 0

co nghiem phan biet f x f x


36/51


Trang 35

a)2 3 4

( ) :2 2

x xH y

x

- +=

-b)

2 3 3( ) :

1x x

H yx

- +=

-c)

2 2 2( ) :

1x x

H yx

+ +=

+

Bai 3. Tm m e tiep tuyen tai iem M bat k thuoc hypebol (H) cat hai ng tiem cantao thanh mot tam giac co dien tch bang S:

a) 2 3( ) : ; 8mxH y S x m

+= =

-

Bai 4. Tm iem M thuoc hypebol (H) tai o tiep tuyen cat cac truc toa o tai cac iem A,B sao cho DOAB vuong can:

a)2 1

( ) :1

x xH y

x

+ +=

-b)

22 5( ) :

2x x

H yx

+=

+c)

2 3 3( ) :

2x x

H yx

+ +=

+

Bai 5. Cho (C): 22 11

x xy

x

- +=

-. Chng minh rang tren ng thang d: y = 7 co 4 iem sao

cho t moi iem co the ke en (C) hai tiep tuyen tao vi nhau mot goc 450.Bai 6. Viet phng trnh tiep tuyen vi ng cong (C) tao vi cac truc toa o mot tam

giac co dien tch S cho trc:a) 1( ) : ; 4C y x S

x= + = b)

3 1 1( ) : ;2

xC y S

x

+= =


37/51


Trang 36

4. HO O TH

Cho ho ng (Cm):y = f(x, m) (m la tham so).M(x0; y0) (Cm) y0 = f(x0, m) (1)

Xem (1) la phng trnh theo an m.

Tuy theo so nghiem cua (1) ta suy ra so o th cua ho (Cm) i qua M. Neu (1) nghiem ung vi moi m th moi o th cua ho (Cm) eu i qua M.Khi o, M c goi la iem co nh cua ho (Cm).

Neu (1) co n nghiem phan biet th co n o th cua ho (Cm) i qua M. Neu (1) vo nghiem th khong co o th nao cua ho (Cm) i qua M.

VAN E 1: Tm iem co nh cua ho o th (Cm):y = f(x, m)Cach 1:

Goi M(x0; y0) la iem co nh (neu co) cua ho (Cm).

M(x0; y0) (Cm), "m y0 = f(x0, m), "m (1)Bien oi (1) ve mot trong cac dang sau:

Dang 1: (1) Am + B = 0, "m Dang 2: (1) 2 0Am Bm C + + = , "m

00

AB

= =

(2a) 000

ABC

=

= =

(2b)

Giai he (2a) hoac (2b) ta tm c toa o (x0; y0) cua iem co nh.

Chu y: Cac he (2a), (2b) la cac he phng trnh co 2 an x0, y0.Cach 2:

Goi M(x0; y0) la iem co nh (neu co) cua ho (Cm).

M(x0; y0) (Cm), "m y0 = f(x0, m), "m (1)at F(m) = f(x0, m) th F(m) = y0 khong oi.

F(m) = 0 (3)Giai (3) tm c x0. Thay x0 vao (1) tm c y0. T osuy ra c cac iem co nh.

Bai 1. Tm cac iem co nh cua ho o th (Cm) co phng trnh sau:a) ( 1) 2 1y m x m= - - + b) 2 2( 2) 3 1y mx m x m= + - - +

c) 3 2( 1) 2 ( 2) 2 1y m x mx m x m= + - - - + + d) 2(1 2 ) (3 1) 5 2y m x m x m= - - - + -

d)3 2

9 9y x mx x m= + - - e)3

( 2) 2y m x mx= - - + f) 4 22 4 1y mx x m= - - + g) 4 2 5y x mx m= + - -

h) ( 1) 2 ( 1, 2)m xy m mx m

- -= - -

-i) 3 1

( 2) 4x m

ym x m

+ -=

+ +

i)2 5 7 2

2 3

x mxy m

mx

- +=

- k)

22 ( 2) ( 0)2

x m x my m

x m

- + + +=

-

l)2

2( 1)

2 2 1

x m x my

x mx m

+ - +=

+ + +m)

2

22 6 4

2 (5 2) 6

x x my

x m x

+ +=

+ + +

Bai 2. Chng minh rang ho o th (Cm) co 3 iem co nh thang hang. Viet phng trnhng thang i qua 3 iem co nh o:a) 3 2( 3) 3( 3) (6 1) 1y m x m x m x m= + - + - + + +


38/51


Trang 37

b) 3 2( 2) 3( 2) 4 2 1y m x m x x m= + - + - + -

c) 3 2( 4) (6 24) 12 7 18y m x m x mx m= - - - - + -

d) 3( 1) (2 1) 1y m x m x m= + - + - +

VAN E 2: Tm iem ma khong co o th nao cua ho o th (C m):y = f(x, m) i quaGoi M(x0; y0) la iem ma khong co o th nao cua ho (Cm) i qua.M(x0; y0) (Cm), "m y0 = f(x0 , m) vo nghiem m (1)

Bien oi (1) ve mot trong cac dang sau:

Dang 1: (1) Am + B = 0 vo nghiem m 00

AB

=

(2a)

Dang 2: (1) 2 0Am Bm C + + = vo nghiem m

2

000

4 0

A BC

A

B AC

= = - 1; k = 2.

b) (Cm):2 2( 1)m x m

yx m

+ -=

-; (L): x > 0; k = 2.

c) (Cm): 4 2 22 1y x mx m= - + + ; (L): y = 1; k = 1.

d) (Cm): 3 2 3( 1) (2 3 2) 2 (2 1)y x m x m m x m m= - + - - + + - ; (L): x = 1, y > 2; k = 2.


40/51


Trang 39

5. TAP HP IEM

Bai toan: Tm tap hp cac iem M(x; y) thoa tnh chat a. Nhan xet: Tm tap hp iem M trong mat phang toa o la tm phng trnh cua taphp iem o.

Dang 1: Tm toa o cua iem M.1) Tm ieu kien (neu co) cua tham so m e ton tai iem M.2) Tnh toa o iem M theo tham so m.Co cac trng hp xay ra:

Trng hp 1: M ( )( )

x f my g m

=

=

Kh tham so m gia x va y, ta co mot he thc gia x, y oc lap vi m co dang:F(x, y) = 0 (goi la phng trnh qu tch)

Trng hp 2: M ( )

( )

x a hang so

y g m

=

=

Khi o iem M nam tren ng thang x = a.

Trng hp 3: M ( )( )

x f my b hang so

= =

Khi o iem M nam tren ng thang y = b.3) Gii han qu tch: Da vao ieu kien (neu co) cua m ( bc 1), ta tm c ieukien cua x hoac y e ton tai iem M(x; y). o la gii han cua qu tch.4) Ket luan: Tap hp cac iem M co phng trnh F(x, y) = 0 (hoac x = a, hoac y = b)vi ieu kien cua x hoac y ( bc 3).

Dang 2: Trong trng hp ta khong the tnh c toa o cua iem M theo tham so m mach thiet lap c mot he thc cha toa o cua M th ta tm cach kh tham so m tronghe thc e tm c he thc dang F(x, y) = 0.

Chu y:Neu bai toan ch hoi : iem M chay tren ng nao th ta ch tm phng trnhF(x, y) = 0 ma khong can tm gii han cua qu tch.

Bai 1. Tm tap hp cac iem ac biet cua ho o th a cho.a) (Pm): 22 ( 2) 2 4y x m x m= - - + - . Tm tap hp cac nh cua (Pm).

b) (Cm): 3 23 2 3 1y x mx x m= - + - - . Tm tap hp cac iem uon cua (Cm).

c) (Cm): 3 22 3(2 1) 6 ( 1) 1y x m x m m x= - + + + + . Tm tap hp cac iem cc ai cua (Cm).d) (Hm):

( 1) 11

m xy

mx

- +=

-. Tm tap hp cac tam oi xng cua (Hm).

e) (Hm):22 3 5

2x mx m

yx

- +=

-. Tm tap hp cac iem cc ai cua (Hm).

Bai 2. Cho (C) va (C). Tm tap hp trung iem cua oan thang.1) Tm m e (C) va (C) cat nhau tai hai iem phan biet A, B.2) Tm tap hp cac trung iem I cua oan thang AB.

a) (C): 3 23 1y x x mx= + + + va (C): 3 22 7y x x= + + .

b) (C): 2 3y x mx= - + va (C): 2y mx= + .

c) (C): 11

xy

x

-=

+va (C): 2 0x y m- + =


41/51


Trang 40

d) (C):2( 2)

1x

yx

-=

-va (C) la ng thang i qua A(0; 3) va co he so goc m.

e) (C):2 4 3

2x x

yx

+ +=

+va (C): 1y mx= + .

Bai 3. Cho (C) va (C).Tm tap hp cac iem.1) Tm m e (C) cat (C) tai 3 iem phan biet A, B, C (trong o xC khong oi).2) Tm tap hp cac trung iem I cua oan thang AB.

a) (C): 3 23y x x= - va (C): y mx= .

b) (C): 3 2 2 22( 1) ( 1)y x m x m x m= - + + + - va (C): 3y mx m= - + .

c) (C): 3 26 9y x x x= - + va (C): y mx= .

d) (C): 2( 2)( 1)y x x= + - va (C) la ng thang i qua C(2; 0) va co he so goc m.Bai 4. Cho (C). Tm tap hp cac iem t o co the ve c hai tiep tuyen cua (C) vuong

goc vi nhau.

a) (C): 1y xx

= + b) (C): 2 11

x xyx+ +=+

Bai 5.a) Cho (C):

21

xy

x

-=

-. Tm tap hp cac iem tren truc tung ma t o co the ke c

tiep tuyen vi (C).b) Cho (C): 3 23 2y x x= - + - . Tm tap hp cac iem tren ng thang y = 2 ma t oco the ke c 3 tiep tuyen vi (C).


42/51


Trang 41

6. HAM SO CO CHA DAU GIA TR TUYET OI

Bai toan: Ve o th cua ham soy = f(x) vif(x) co cha dau gia tr tuyet oi.Cach 1: Khao sat s bien thien va ve o th.

Xet dau bieu thc co cha dau gia tr tuyet oi.

Chia mien xac nh thanh nhieu khoang, trong moi khoang ta bo dau gia tr tuyet oi. Ve o th ham so tng ng trong cac khoang cua mien xac nh.

Cach 2: Thc hien cac phep bien oi o th.Dang 1: Ve o th ham so ( )y f x= .

o th (C) cua ham so ( )y f x= co the c suy t o th (C) cua ham soy = f(x)nh sau:

+ Gi nguyen phan o th (C) pha tren truc hoanh.+ Lay oi xng phan o th cua (C) pha di truc hoanh qua truc hoanh.+ o th (C) la hp cua hai phan tren.

Dang 2: Ve o th cua ham so ( )y f x= .o th (C) cua ham so ( )y f x= co the c suy t o th (C) cua ham soy = f(x)

nh sau:+ Gi nguyen phan o th (C) ben phai truc tung, bo phan ben trai truc tung.+ Lay oi xng phan ben phai truc tung qua truc tung.+ o th (C) la hp cua hai phan tren.

Bai 1. Khao sat s bien thien va ve o th (C). T o suy ra o th C). Dung o th (C)bien luan so nghiem cua phng trnh (1):

a) (C): 3 23 6y x x= - - ; (C): 3 23 6y x x= - - ; 3 23 6x x m- - = (1)

b) (C): 4 22 3y x x= - - ; (C): 4 22 3y x x= - - ; 4 22 3x x m- - = (1)

c) (C):22 5 2

1x x

yx

+ -=

+; (C):

22 5 21

x xy

x

+ -=

+;

22 5 21

x xm

x

+ -=

+(1)


43/51


Trang 42

d) (C):2 1

2x x

yx

- -=

-; (C):

2 12

x xy

x

- -=

-;

2 12

x xm

x

- -=

-(1)

e) (C): 2 22

xy

x

-=

-; (C): 2 2

2x

yx

-=

-; 2 2

2x

mx

-=

-(1)

Bai 2. Khao sat s bien thien va ve o th (C). T o suy ra o th C). Dung o th (C)bien luan so nghiem cua phng trnh (1):a) (C): 3 22 9 12 4y x x x= - + - ; (C):

3 22 9 12 4y x x x= - + - ;3 22 9 12x x x m- + =

b) (C): 21

xy

x=

-; (C): 2

1x

yx

=-

; ( 2). 0m x m- - = (1)

c) (C):2 4 5

2x x

yx

+ +=

+; (C):

2 4 52

x xy

x

+ +=

+;

2 4 52

x xm

x

+ +=

+(1)

Bai 3. Khao sat s bien thien va ve o th (C). T o suy ra o th C ). Dung o th (C),tm m e phng trnh (1) co k nghiem phan biet:

a) (C): 4 22 1y x x= - - ; (C): 4 22 1y x x= - - ; 4 2 22 1 logx x m- - = ; k = 6.b) (C): 3 26 9y x x x= - + ; (C):

3 26 9y x x x= - + ;3 26 9 3 0x x x m- + - + = ; k = 6.

c) (C):22 5 2

1x x

yx

+ -=

+; (C):

22 5 21

x xy

x

+ -=

+;

22 5 21

x xm

x

+ -=

+; k = 4.

d) (C):4

2 532 2

xy x= - + ; (C):

42 53

2 2x

y x= - + ;4

2 253 22 2

xx m m- + = - ; k = 8.


44/51


Trang 43

7. IEM AC BIET TREN O TH CUA HAM SO

VAN E 1: Tm iem tren o th (C):y = f(x) co toa o nguyen

Tm cac iem tren o th ham so hu t( )( )

P xy

Q x= co toa o la nhng so nguyen:

Phan tch ( )( )

P xyQ x

= thanh dang ( )( )ay A x

Q x= + , vi A(x) la a thc, a la so nguyen.

Khi o xy

Q(x) la c so cua a. T o ta tm cac gia tr x nguyen e Q(x) la

c so cua a.Th lai cac gia tr tm c va ket luan.

Bai 1. Tm cac iem tren o th (C) cua ham so co toa o nguyen:a)

2

1

x

y x

+

= + b)

10

2

x

y x

-

= + c)

2

2

x

y x

+

= -

d)2 1

2x x

yx

+ +=

+e)

2 21

x xy

x

+=

+f) 41

1y x

x= + +

-

Bai 2. Tm cac iem tren o th (C) cua ham so co toa o nguyen:a) 2 2( 1) 4y x y x y x= + + + + b) 22 4( 1) 6y x y x y x= + + - +

VAN E 2: Tm cap iem tren o th (C): y = f(x)

oi xng qua ng thang d:y = ax + bC s cua phng phap: A, B oi xng nhau qua dd la trung trc cua oan ABPhng trnh ng thang D vuong goc vi d: y = ax = b co dang:

D:1

y x ma

= - +

Phng trnh hoanh o giao iem cua D va (C):

f(x) =1

x ma

- + (1)

Tm ieu kien cua m eD cat (C) tai 2 iemphan biet A, B. Khi o xA, xB la cac nghiem cua (1).

Tm toa o trung iem I cua AB.T ieu kien: A, B oi xng qua d I d, ta tm

c m xA, xByA, yBA, B.

Chu y:A, B oi xng nhau qua truc hoanh A B

A B

x x

y y

= = -

A, B oi xng nhau qua truc tung A B

A B

x x

y y

= -

=

A, B oi xng nhau qua ng thang y = b

2

A B

A B

x x

y y b

=

+ =

A, B oi xng nhau qua ng thang x = a 2A B

A B

x x a

y y

+ = =

(d)(C)

(D)

BA I


45/51


Trang 44

Bai 1. Tm tren o th (C) cua ham so hai iem oi xng nhau qua ng thang d:a) 3( ) : ; : 2 0C y x x d x y= + + = b)

4( ) : ; : 2 6 0

2x

C y d x yx

+= - - =

-

c)2

( ) : ; : 11

xC y d y x

x= = -

-d)

2 1( ) : ; : 11

x xC y d y x

x

+ -= = -

-

Bai 2. Cho o th (C) va ng thang d. Viet phng trnh o th (C) oi xng vi (C) quang thang d:a) 3 2( ) : 3 5 10 2; : 2C y x x x d x= - + - = - b)

22 3 7( ) : ; : 21

x xC y d x

x

- += =

-

c)2 2( ) : ; : 2

2x x

C y d yx

+ -= =

-d)

22 5 3( ) : ; : 11

x xC y d y

x

+ -= = -

-

Bai 3. Tm m e tren o th (C) co mot cap iem oi xng nhau qua ng thang d:a) 3 2 2( ) : 3 2 ; :C y mx x x m d Ox= + + +

VAN E 3: Tm cap iem tren o th (C): y = f(x) oi xng qua iem I(a; b)C s cua phng phap: A, B oi xng nhau qua II la trung iem cua AB.

Phng trnh ng thang d qua I(a; b),co he so goc k co dang: ( )y k x a b= - + .Phng trnh hoanh o giao iem cua (C) va d:

f(x) = ( )k x a b- + (1)Tm ieu kien e d cat (C) tai 2 iem phan biet

A, B. khi o xA, xB la 2 nghiem cua (1).T ieu kien: A, B oi xng qua II la trung iem cua AB, ta tm c kxA, xB.

Chu y: A, B oi xng qua goc toa o O A B

A B

x xy y

= -= -

Bai 1. Tm tren o th (C) cua ham so hai iem oi xng nhau qua iem I:a) 3 2( ) : 4 2; (2;4)C y x x x I = - + + b)

2 2 5( ) : ; 0;1 2

x xC y I

x

+ +=

-

c) 3 2( ) : 3 2 1; (0;0)C y x x x I O= - - + d) 4( ) : ; (0;0)1

xC y I O

x

+=

+

e) 3 4( ) : ; (1;1)2 1xC y Ix+= - e) ( )2

2 5 1( ) : ; 2; 51x xC y Ix- += - -+

Bai 2. Cho o th (C) va iem I. Viet phng trnh o th (C) oi xng vi (C) qua iem I:a) 3 2( ) : 2 3 5 1; (1;2)C y x x x I = + + + b)

2( 1)( ) : ; (1;1)2

xC y I

x

-=

-

c)2 1( ) : ; (2;1)

1x x

C y Ix

- +=

-d)

3 22 5 1( ) : ; (2;1)2 3

x x xC y I

x

- - +=

-

Bai 3. Tm m e tren o th (C) co mot cap iem oi xng nhau qua iem:a) 3 2 2 2( ) : 3 3( 1) 1 ; (0;0)C y x mx m x m I O= - + - + -

b) 3 2( ) : 7 3; (0;0)C y x mx x I O= + + +

c) 3 2( ) : 9 4; (0;0)C y x mx x I O= + + + d)2 2 22

( ) : ; (0;0)1

x m x mC y I O

x

+ +=

+

ABI


46/51


Trang 45

VAN E 4: Khoang cachKien thc c ban:

1) Khoang cach gia hai iem A, B: AB = 2 2( ) ( )B A B Ax x y y- + -

2) Khoang cach t iem M(x0; y0) en ng thang D: ax + by + c = 0:

d(M, D) = 0 02 2

ax by c

a b

+ +

+

3) Dien tch tam giac ABC:

S = ( )22 21 1. .sin . .

2 2AB AC A AB AC AB AC = -

uuur uuur

Bai 1. Cho o th (C) va iem A. Tm iem M tren (C) sao cho AM nho nhat. Chng minhrang khi AM nho nhat th ng thang AM vuong goc vi tiep tuyen cua (C) tai M.a) 2( ) : 1; (0;0)C y x A O= - b) 2( ) : ; (3;0)C y x A=

c) 2( ) : 2 1; (9;1)C y x A= + Bai 2. Cho o th (C) va ng thang d. Tm iem M tren (C) sao cho khoang cach t M

en d la nho nhat.

a) 4 2( ) : 2 3 2 1; : 2 1C y x x x d y x= - + + = - b)2 4 5

( ) : ; : 3 62

x xC y d y x

x

+ += = - -

+

c) 2( ) : ; : 2( 1)C y x x d y x= - = + d)1

( ) : ; : 2 31

xC y d y x

x

+= = - +

-

Bai 3. Tm cac iem M thuoc o th (C) sao cho d(M,Ox) = k.d(M,Oy) vi k cho trc.a)

2( ) : ; 12

xC y kx

+= =- b)

2 1( ) : ; 11

x xC y kx

+ -= =-

c)2 1( ) : ; 2

1x x

C y kx

+ -= =

-d)

2 2 2( ) : ; 21

x xC y k

x

+ += =

+

Bai 4. Tm cac iem M thuoc hypebol (H) sao cho tong cac khoang cach t o en haitiem can la nho nhat.

a)2

( ) :2

xH y

x

+=

-b)

2 1( ) :

1x

H yx

-=

+c)

4 9( ) :

3x

H yx

-=

-

d)2 2( ) :

3

x xH y

x

+ -=

-

e)2 1( ) :

2

x xH y

x

- +=

-

f)2 3 3( ) :

2

x xH y

x

+ +=

+

Bai 5. Tm cac iem M thuoc hypebol (H) sao cho tong cac khoang cach t o en hai tructoa o la nho nhat.

a) 1( ) :1

xH y

x

-=

+b) 2 1( ) :

2x

H yx

+=

-c) 4 9( ) :

3x

H yx

-=

-

d)2 11

( ) :1

x xH y

x

+ -=

-e)

2 3( ) :

2x

H yx

-=

-f)

2 6( ) :

3x x

H yx

+ -=

-

Bai 6. Tm cac iem M thuoc hypebol (H) sao cho khoang cach t o en giao iem cuahai tiem can la nho nhat.

a)2

2 2( ) : 1x xH y x+ += - b)2

1( ) : ; 11x xH y xx- += >-

Bai 7. Cho hypebol (H). Tm hai iem A, B thuoc hai nhanh khac nhau cua (H) sao cho odai AB la nho nhat.


47/51


Trang 46

a) 1( ) :1

xH y

x

-=

+b) 2 3( ) :

2x

H yx

+=

-c) 4 9( ) :

3x

H yx

-=

-

d)1

( ) : 2 1H y xx

= + + e)2 3 3

( ) :1

x xH y

x

- +=

-f)

2 2 5( ) :

1x x

H yx

- +=

-

Bai 8. Cho (C) va ng thang d. Tm m e d cat (C) tai 2 iem A, B sao cho o dai AB lanho nhat.a)

2 6 4( ) : ; :1

x xH y d y k

x

+ -= =

+b) 1( ) : ; : 2 0

1x

H y d x y mx

+= - + =

-


48/51


Trang 47

Bai 1. Cho ham so: 3 2 4,y x ax= + - a la tham so.a) Khao sat va ve o th vi a = 3.b) Tm cac gia tr cua tham so a e phng trnh sau co nghiem duy nhat:

3 2 4 0x ax+ - = S: b) a < 3.

Bai 2. a) Khao sat va ve o th ham so: 3 26 9 1y x x x= - + - .b) T mot iem bat ky tren ng thang x = 2 ta ke c bao nhieu tiep tuyen ti oth cua ham so?

S: b) mot tiep tuyen.

Bai 3. Cho ham so:3

3 (1)y x x= - a) Khao sat s bien thien va ve o th cua ham so (1).b) Chng minh rang m khi thay oi, ng thang d cho bi phng trnh:

( 1) 2y m x= + + luon cat o th ham so (1) tai mot iem A co nh. Hay xac nh cacgia tr cua m e ng thang d cat o th ham so (1) tai 3 iem A, B, C khac nhau saocho tiep tuyen vi o th tai B va C vuong goc vi nhau.

S: b) 2( 1; 2); 1 23

A m- = - +

Bai 4. a) Khao sat va ve o th ham so: 4 22 1 (1)y x x= - - b) Vi nhng gia tr nao cua m th phng trnh sau co 6 nghiem phan biet.

4 242 1 log (2)x x m- - =

S: b) 4 < m < 16.

Bai 5. Cho ham so: 4 25 4 (1)y x x= - + a) Khao sat va ve o th (C) cua ham so.b) Tm ieu kien cua tham so m e ng thang y = m cat o th (C) cua ham so tai 4iem phan biet.c) Tm m sao cho o th (C) cua ham so chan tren ng thang y = m ba oan thang coo dai bang nhau.

S: b) 9 44

m- < < c) 74

m =

Bai 6. Cho ham so: 4 21 32 2

y x mx= - + (1)

a) Khao sat va ve o th (C) cua ham so khi m = 3.

b) Viet phng trnh tiep tuyen i qua 30;2

A

tiep xuc vi (C).

c) Xac nh m e ham so (1) co cc tieu ma khong co cc ai.S: b) 3 3; 2 2

2 2y y x= = + c) m 0.

VIII. ON TAP KHAO SAT HAM SO


49/51


Trang 48

Bai 7. Cho ham so: 3 4 ( )1

xy H

x

+=

-

a) Khao sat va ve o th ham so.b) Vi gia tr nao cua a, ng thang y = ax + 3 khong cat o th (H)?c) Qua iem M(2 ; 3) viet phng trnh tiep vi o th (H).

S: b) 28 < a 0 c) y = 28x + 59.

Bai 8. a) Khao sat va ve o th 2 ( )1

xy C

x

-=

-.

b) Tm tat ca nhng iem tren o th (C) cach eu hai iem A(0; 0) va B(2; 2).S: b) (2 ; 0), (0 ; 2).

Bai 9. Cho ham so: 12 ( )y x C x

= - +

a) Khao sat va ve o th (C).b) Tm tren (C) cac iem cach eu hai truc toa o.

c) Tm k e ng thang y = k cat (C) tai hai iem ma tai o hai tiep tuyen vi (C)vuong goc vi nhau.

S: b)1 1;2 2

M

c) 2 5.k= -

Bai 10. Cho ham so: 2 2( 1) 4 4 2( 1)

x m x m my

x m

- + + - -=

- -

a) Khao sat va ve o th vi m = 2.b) Tm cac gia tr cua m e ham so xac nh va ong bien tren khoang (0 ; +)

S: b) 2 3 37 2m-

Bai 11. a) Khao sat s bien thien va ve o th (C) cua ham so: 2 2 21

x xy

x

+ +=

+.

b) Goi I la tam oi xng cua o th (C) va M la mot iem tren (C). Tiep tuyen tai Mvi (C) cat hai ng tiem can tai A va B. Chng minh rang M la trung iem cua oanAB va dien tch tam giac IAB khong phu thuoc vao v tr iem M tren (C).

S: b) 2 2.IABS =

Bai 12. Cho ham so:2 2 2 1

1 ( )1 1

x x

y x C x x

+ +

= = + ++ + a) Khao sat va ve o th (C).b) Tm tren o th ham so a cho cac iem sao cho tiep tuyen tai o vuong goc vitiem can xien cua no.

S: b) 1 22 3 2 2 3 21 ; ; 1 ;

2 2 2 2M M

- + - - -

Bai 13. Cho ham so: 2 ( 1) 1 ( )mx m x mxy Cx m+ + - +

=-

a) Khao sat va ve o th (C) cua ham so ng vi m = 2.b) Chng minh rang tch cac khoang cach t mot iem tuy y thuoc o th (C) (vi m = 2 cau tren) ti hai ng tiem can luon bang mot hang so.


50/51


Trang 49

c) Vi gia tr nao cua m th ham so a cho co cc ai, cc tieu, ong thi gia tr ccai va gia tr cc tieu cung dau.

S: b)9 2

2c) 3 2 3 3 2 3m hay m< - - > - +

Bai 14.

a) Khao sat va ve o th ham so:

2 4 1

2

x x

y x

+ +

= + b) Tm cac iem tren o th co khoang cach en ng thang (D) : y + 3x + 6 = 0 lanho nhat.

S: b) 1 23 5 5 5

; ; ;2 2 2 2

M M

- - -

.

Bai 15. Cho ham so: 22 21

x mxy

x

+ -=

-vi m la tham so.

a) Xac nh m e tam giac tao bi hai truc toa o va ng tiem can xien cua o thcua ham so tren co dien tch bang 4.b) Khao sat va ve o th ham so tren khi m = 3.

S: a) m = 6 hay m = 2.

Bai 16. Cho ham so: 2 1x xyx

+ += .

a) Khao sat s bien thien va ve o th cua ham so tren.b) Xac nh m sao cho phng trnh sau co nghiem:

4 3 2( 1) 3 ( 1) 1 0t m t t m t - - + - - + =

S: b)3 7

.2 2m hay m -

Bai 17. Cho ham so: 3 2 2 2 23 3(1 ) (1)y x mx m m m= - + + - + - (m la tham so)a) Khao sat s bien thien va ve o th ham so (1) khi m = 1.

b) Tm k e phng trnh 3 2 3 23 3 0x x k k - + + - = co 3 nghiem phan biet.c) Viet phng trnh ng thang i qua hai iem cc tr cua o th ham so (1).

S: b) 1 3; 0; 2;k k k- < < c) 22y x m m= - +

Bai 18. Cho ham so: 4 2 2( 9) 10y mx m x= + - + (1) (m la tham so)a) Khao sat s bien thien va ve o th cua ham so (1) khi m = 1.b) Tm m e ham so (1) co ba iem cc tr.

S: b) 3 0 3.m hay m< - < <

Bai 19. Cho ham so: 2(2 1) (1)1

m x my

x

- -=

-(m la tham so)

a) Khao sat s bien thien va ve o th (C) cua ham so (1) ng vi m = 1.b) Tnh dien tch hnh phang gii han bi ng cong (C) va hai truc toa o.c) Tm m e o th cua ham so (1) tiep xuc vi ng thang y = x.

S: b) 41 4 ln3

S= + c) m 1.


51/51


Bai 20. Cho ham so: 21

mx x my

x

+ +=

-(1) (m la tham so)

a) Khao sat s bien thien va ve o th ham so (1) khi m = 1.b) Tm m e o th ham so (1) cat truc hoanh tai hai iem phan biet va hai iem o cohoanh o dng.

S: b) 1 0.2

m- < <

Bai 21. Cho ham so: 3 23y x x m= - + (1) (m la tham so)a) Tm m e ham so (1) co hai iem phan biet oi xng nhau qua goc toa o.b) Khao sat s bien thien va ve o th ham so (1) khi m = 2.

S: a) m > 0.

Bai 22. a) Khao sat s bien thien va ve o th cua ham so 2 2 4 (1)2

x xy

x

- +=

-

b) Tm m e ng thang dm: y = mx + 2 2m cat o th cua ham so (1) tai hai iemphan biet.

S: b) m > 1.

Bai 23. Cho ham so: 2 3 32( 1)

x xy

x

- + -=

-(1)

a) Khao sat s bien thien va ve o th cua ham so (1).b) Tm m e ng thang y = m cat o th tai 2 iem A, B sao cho AB = 1.

S: b) 1 5

2

m

= .

Bai 24. Cho ham so: 3 21 2 3 (1)3

y x x x= - + co o th (C)

a) Khao sat s bien thien va ve o th cua ham so (1).b) Viet phng trnh tiep tuyen D cua (C) tai iem uon va chng minh rang D la tieptuyen cua (C) co he so goc nho nhat.

S: b) 8: ; 1.3

y x k = - + = -D

Bai 25. Cho ham so: 3 23 9 1 (1)y x mx x= - + + (vi m la tham so)a) Khao sat s bien thien va ve o th cua ham so (1) khi m = 2.b) Tm m e iem uon cua o th ham so (1) thuoc ng thang y = x + 1.

S: b) m = 0 hay m = 2 hay m = 2.

Documents

com BaiTapGiaiTich12 KhaoSatHamSo TranSiTung