Thiết kế bài giảng đại số 10 nâng cao (tập 2) phần 1

TRAN VINH

rturso
TAP HAI

10
//

-2

-1 0

NHA XUAT BAN HA NOI



TRAN VINH

THIET KE BAI GIANG
j^"

[)AI SO
NANG CAO - TAP HAI

NHA XUAT BAN HA NOI



CbirdNq IV

BAT DANG THlTC

VA BAT PHl/OlNC TRINH
PHAN1

I.

NOI DUNG
Ndi dung chinh cua chuong IV :

Bat phuong trinh : Bat ding thiic ; Dai cuong ve bat phuong trinh ; Bat
phuong trinh va he bat phuong trinh mdt an; Dau cua nhi thiic bae nhat ; Bat
phuong trinh va he bat phuong trinh bae nha't hai dn; Dau cua tam thiic bae hai;
Bat phirong trinh bae hai; Mdt sd phuong trinh va bat phuong trinh quy ve bae
hai.

B a t d a n g thaiti
On tap ve khai niem bat dang thiic; Bat dang thuc ve trung binh cdng va
trung binh nhan (bat ding thiic Cd - si); Bat dang thiic chiia gia tri tuyet ddi.
Ndi dung chinh cua phan nay la xay dung quan he thii tu tren tap hgp sd
thuc. Sach trinh bay mdt each het siic co ban, he thdng ve bat ding thiic. Trong
phan nay ciing dua ra cac tinh chat, quy tdc ve bat ding thiic, tir dd giiip hgc
sinh chiing minh dugc cac bat ding thuc don gian. Dac biet trong phan nay cd
dua vao hai bat ding thiic : Bat ding thiic v6 trung binh cdng va trung binh
nhan; bat ding thiic cd gia tri tuyet ddi

B a t p]»<«*iig t r i n l i
V6 mat hinh thiic, bat phuong trinh dugc trinh bay co ban nhu' phuong
trinh, nhung chi khac d day da xay dung quan he thii tu tren tap hgp sd thuc.
Ndi dung co ban la trinh bay ve bat phuong trinh mdt an ma d do cung dua ra
cac quy tic co ban de tim nghiem ciia chiing. Dac biet la ndi dung ve kinh dien
la dau cua nhi thiic bae nhat va dau tam thuc bae hai, ap dung chung trong viec

3


giai bat phuong trinh bae nhat va bat phuong trinh bae hai. Trong chuong nay
cung dua them khai niem ve bat phuong trinh bae hai hai in, d dd cd dang dap
ciia bai loan kinh te. bai toan tdi uu.
II.

MUC TIEU

1.

Kien thufc
Nim dugc loan bd kien thiic co ban trong chuong da neu tren, cu the :
Hieu khai niem bat ding thiic; chiing minh bat ding thirc.

Hieu y nghia cac quy tic va tinh chat cua bat ding thiic va van dung
dugc trong viec chirng minh bat ding thirc.
Nim dugc mdt each chic chin ve khai niem bat phuong trinh, tap
nghiem va quan he tuong duong ciia cac bat phuong trinh.
Nim dugc cac kien thuc co ban nha't vd dau cua nhi thirc bae nhat va
tam thiic bae hai. van dung linh hoat trong viec giai bat phuong trinh
Nim dugc cac khai niem Ve bat phuong trinh bae nha't hai in. tap
nghiem ciia chiing va van dung trong giai bai toan kinh te.
2.

KT nang

Chiing minh thanh thao cac bat ding thirc, van dung linh hoat cac quy
tic: chuyen ve. nhan hoac chia hai \6 cua mdt bat ding thuc vdi mdt sd.

Giai mdt each thanh thao cac bat phuong trinh bae nhat xa bat phuong
trinh bae hai.
•DBiet xac dinh tap nghiem cua bat phuong trinh bae nhat hai in, tir do
manh nha giai cac bai toan kinh te rat don gian.
3.

Thai do

•DTir giac, tich cue. ddc lap va chu ddng phat hien cQng nhu linh hdi kien
thu'c trong qua trinh hoat ddng.
Cin than chinh xac trong lap luan va tinh toan.


PHAN 2
CAC B A I SOAKT

§1. Bat dang" thdc va cluitng m i n h b a t
dang^ thu'c (tiet 1, 2, 3)
I.

MUC TIEU

1.

Kien thiic

HS nim dugc :
Cac khai niem ve bat ding thirc.
Cac tinh chat ciia bat ding thiic.
Cac bat ding thiic co ban va cac tinh chat cua nd.

He thdng dugc cac bat ding thuc, tir dd hinh thanh cac phuong phap
chiing minh cac bat ding thuc.
Van' dung cac bat ding thiic Cd-si, bat ding thiic cd chua dau gia tri
tuyet ddi de giai cac bai tap cd lien quan.
Biet tim gia tri Idn nhat va nhd nhat ciia mdt ham sd, mot bieu thiic dua
vao bat ding thiic.
2.

KT nang
HS phai chiing minh dugc cac bat ding thiic don gian.

Van dung thanh thao cac tinh chat cua bat ding thirc de bien ddi. tir dd
giai dugc cac bai toan \'6 chiing minh bat ding thiic, tim gia tri Idn nhat, nhd
nhat cua ham sd, ciia mdt bieu thiic.
3.

Thai do
Tu giac, tich cue trong hgc tap.

Biet phan biet rd cac khai niem co ban, cac tinh chat va van dung trong tirng
trudng hgfp cu the.
Tu duy cac van di cua toan hgc mdt each logic va he thdng, budc diu
CO tu duy cue tri trong qua trinh sang tao.


II.

CHUAN BI CUA GV VA HS

1.

Chuan bi cua GV

D6 dat cau hdi cho HS, trong qua trinh thao tac day hgc GV cd the
chuin bi mdt sd kien thiic ma HS da hgc d ldp 8 v6 bat ding thiic
Chuin bi phan mau \h. mdt sd cdng cu khac.
2.

Chuan bi ciia HS
Can dn lai mot sd kien thiic da hgc d ldp dudi.

III. PHAN PHOI THCH LUONG
Bai nay chia lam 3 tiet :
Tiet 1 : Tit ddu den het phdn 2.
Tiet 2 : Tit phdn 3.
Tiet edn Iqi: chda bdi tap.
IV

TIEN TRINH DAY HOC

A.

OAT VAN OE
Cau hdi 1
So sanh cac sd sau:
a) 2007^°°^ va 2006^°°^
b) 5 + V3 va (1 + V3)2
Cau hoi 2
Nhirng ket luan sau day, ket luan nao dung?
(a) x~ + X + 1 > 0 vdi mgi x

G M.

- (b) Vi 3 > 2 nen 3a > 2a Va ^ 0.
( c ) x - - l > 0 Vx G R.
(d) 3 > 2 va a > b nen 3a > 2b.
GV : Chi co (a) Id diing.


B.

BAI Mdi
HOAT DONG 1

1. On Tap va bo sung tinh chat ciia bat ding thijfc
• GV neu van de:
HI. Hay neu khai niem ve bat ding thiic.

_

H2. The nao la chiing minh bat ding thuc?
H3. a +h < -2 cd phai la bat dang thiic hay khdng? neu phai thi day la bat dang
thiic diing hay sai?
• GV neu dinh nghia:
Cdc menh de dang a < b hoac "a > b" "a > b" "a < b"
dugc ggi la nhiing bdt ddng thdc.
Hoat ddng cua GV
Cau hoi 1
Dien vao chd trd'ng:
a < b c ^ a - b < ....

a-b<0<:^a...b.
Cau hoi 2
dien vao chd trd'ng
a - b - 1 < 0 <:^ a < ...

Hoat dong ciia HS
Ggi y tra 161 cau hoi 1
aa-b<0<^aGgi y tra idi cau hoi 2
a - b - l < O c : > a < b + 1.

Tinh chat ciia bat ding thiic
GV: Cho HS dgc vd xem xet trang 104 SGK. Sau do chia HS thanh 4 nhom,
mdi nhom thao luan vd cii dai dien len bdng thuc hien thao tdc dien vao
chS trdng.


Nhom 1. Dien ddu > hoac < vao cho tiring.
Tinh chdt
Dieu kien

Ten goi

Noi dung
a
+ c...b

+c

oO

a < b c^ ac ... be

c<0

a < b o ac ... be
a < b va c < d ^> a + c ... b + d

(3 > 0, c > 0

a
... bd

Cdng hai ve cua bat ding
thuc vdi mot sd
Nhan hai ve ciia bat ding
thire vdi mdt sd
Cdng hai bat ding thuc
cung chi6u
Nhan hai bat ding thiic
cung chieu

a > b c^ }fa..Mb
Nhom 2. Dien ddu > hoac < vao cho trd'ng.
Tinh chdt
Ten goi
Dieu kien

n nguyen
duong
a>0

Noi dung
a >b>0=>

a"...b"

a < b <=> yla...^b
a < b c^ yfa...yfb

Nang hai ve ciia bat ding
thiic len mot luy thira
Khai can hai ve cua mdt bat
ding thiic

Nhom 3. Dien => hoac c^ vao chS trdng.
Tinh chdt
Dieu kien

Ten goi
Noi dung
a
Cdng hai ve cua bat ding
thirc vdi mdt so


oO

a
c<0

a <b ... ac> be
a
a>0,c>0

Nhan hai ve ciia bat ding
thiic vdi mdt sd

+ c
a
Cdng hai bat ding thuc
cung chieu
Nhan hai bat ding thirc
cung chifiu

a Nhom 4. Dien => hoac o vao cho trd'ng.

Tinh chdt
Dieu kien
n nguyen
duong
a>Q>

Ten gpi

Noi dung
a>h>0

...a" > b"

a < b ... yfa < \/b
a
Nang hai ve cua bat ding
thiic len mdt luy thira
Khai can hai ve ciia mdt bat
ding thii'c

Sau dd GV neu tdm tit cac tinh chat va he qua cua cac bat ding thiic trong SGK.
+ Cdc tinh chdt:
a > b va b > c ^> a > c;

a > b C:> a + c > b + c

Neu c > 0 thi a > b <:^ a c > be ;

Neu c < 0 thi a > b <» ac < be.

+ Cdc he qua:
a > b va c > d = > a + c > b + d ;

a + c>b <:^a>b-c;

a > b > 0 va c > d > 0 ^

a > b > 0 va

a>b>0=5>^/a>^y^;

a.c> b.d ;

UGN*

^

a">b";

a>h => yfa > ifb

• GV neu vi du 1 trong SGK va cho HS lam, ggi y bing cac cau hdi sau:


HI. Binh phuong hai ve cua mdt bat ding thirc, ta dugc bat ding thiic ciing chi6u
Diing hay sai?
H2. Neu sai can bd sung dieu kien gi de dugc khing dinh diing?
H3. Gia sit v2 + V3 > 3. hay binh phuong hai ve va so sanh.
• GV neu quy udc trong SGK.
• Neu vi du 2 trong SGK va cho HS lam, ggi y bing cac cau hdi sau:
HI. Hay chuyen ve va dua bat ding thiic ve dang : f(x) > 0
H2. Hay chiing minh f(x) > 0 la bat ding thirc diing.
• Neu vi du 3 trong SGK va cho HS lam, ggi y bing cac cau hdi sau:
HI. Hay neu mdi quan he giira cac canh trong tam giac.

2

2

2

H2. Giai thich vi sao a > a - (b - c)
H3. Giai thich vi sao (a - b + c)(a + b - c) = a^

-(b-c)^

H4. Hay lam tuong tu va chiing minh bat ding thiic da cho.
2. Bat dang thurc ve gia tri tuyet ddi.
• GV neu cac bat ding thiic trong SGK
-\a\|x| < a <=> - a < X < a vdi a > 0.
\x\>a <^ x <-a hoac .v > a vaia >0.
• GV neu bat ding thirc quan trgng sau:
\a\ - \b\ <\a + b\< \a\ + \b\ (vdi mgi a, b e R).
GV cho HS chiing minh bat dang thiic tren (ke ca |H1| bing HD sau)

10


Hoat dong cua H S

Hoat dong cua G V
Cau hoi 1
Hay chiing minh bat dang
thiic :

\a + b\< \a\ + \b\
bang each binh phirang hai
ve.

Ggi y tra 161 cau hoi 1
\a + b\ < \a\ + \b\
c^ (a + b)^ < a^ + 2\ab\ + b^
<=> a^ + 2ab + b^ < a^ + 2\ab\ + b^
o

Cau hoi 2
Hay chimg minh bat dang
thiic :
\a\ - \b\ <\a + b\
b i n g each binh phirang hai
ve.

ab < \ab\

Ggi y tra 161 cau hoi 2
Ifir + ^l > \a\ - \b\
<:^ (a + b)^> a^-2\ab\
o

+ b^

a^ + 2ab + b^ > a^ - 2\ab\ + h^

Ci> ab > - \ab\
Cau hoi 3

Hay chiing minh bat dang
thiic :
lai - \b\ <\a + b\
b i n g each sir dung cau hdi 1

Ggi y tra 161 cau hdi 3
Ta cd |a| = |a + Z) - Z?| < |a + Z?| + \-b\
= \a + b\ + \b\
Chuyen ve ta dugc DPCM.

HOAT DONG 2
2. Bat d^ng thiifc trung binh cong va trung binh nhan

a) Do! vdi hai so khong am
• GV neu dinh li.
Vdi mgi a > 0, b > 0 ta cd
a+b

i—r
> yjab

2
Ding thirc xay ra khi va chi khi a = b.
11


Sau dd GV neu cac cau hdi
HI. Hay phat bieu dinh li bing Idi.
De chung minh dinh li, GV neu cac cau hdi sau:
H2. Dien cac dau >. < >. < vao chd trdng sau:

^ ' ^ = - - ( a + b- 2 ^ )
2
2

= - - ( ^ - V^)2...0,
2

H3. Hay ket luan va chi ra trudng hgp dau bing xay ra.
H4. Van dung dinh li hay chii'ng minh I tan x + cot x 1> 2
Thuc hien H2
GV cho HS doc va hieu noi dung H2
GV treo hinh 4.1. Sau do thuc hien theo cac thao tac sau:
GS-^ thao tdc trong 3 phiit.
Hoat dong cua GV
Cau hoi 1
Hay xac dinh gdc ACB
Cau hdi 2
Hay tinh OD.
Cau hdi 3
Hay tinh CH.
Cau hdi 4
Hay so sanh OD va CH, tCr dd
riit ra bat dang thiic.

Hoat dong cua HS
Ggi y tra 161 cau hdi 1
ACB =90°
Ggi y tra 161 cau hdi 2

0D=^+^

2
Ggi y tra 161 cau hoi 3
CH =
^
Ggi y tra 161 cau hdi 4
OD>CH

tir do t a c d

''^^>V^
2

• GV neu vi du 4 va hudng din HS lam theo cac cau hdi sau:
HI. Hay phan tich \'e trai thanh tong ciia nhCing sd cd dang nghich dao ciia nhau.
H2. Hay ap dung bat ding thuc Cd-si.
12


• GV phat bieu he qua bing Idi:
HE QUA
NcAt hai sd duang thay doi nhung co tong khong ddi thi tich ciia
chiing lini nhdt khi hai sd do bdng nhan. Nen hai sd duang thay doi
nhung cd ticli khong ddi thi tong ciia chiing nho nhdt khi hai so do
bang nhau.
Sau dd hudng dan HS chiing minh theo cac cau hdi sau day:
S
rHl. Chiing minh — > Jxy
c

2

H2. xy Idn nhat khi nao?
H3. Chung minh x + y > 2 V ^
H4. x+ y nhd nhat khi nao?
H5. Ap dung he qua hay tim gia tri nhd nhat cua bi6u thiic ciia bieu thirc :

• GV neu iing dung :
Trong tdt cd cdc hinh chif nhdt cd cimg chii vi, hinh viiong cd dien
tich lan nhdt. Trong td'i cd cdc hinh chit nhdt cd ciing dien tich,
hinh viidng cd chu vi nho nhdt.
• GV neu vi du 5 va hudng din HS thuc hien theo cac cau hdi sau:
HI. Chiing minh ring f(x) > 2V3
H2. Dau bing xay ra khi nao?
b) Ddi vdi ba sd khdng am
• GV neu dinh li

13


a+b+c

^Jabc <» a = b- c

(vdi mgi a, b, c > 0).
Sau dd GV dua ra cac cau hdi sau:
HI. Hay phat bieu dinh li bing Idi.
H2. Neu bd di dieu kien ba sd khdng am thi dinh li edn diing hay khdng? hay neu
mdt vi du.
• GV neu vi du 6.
Sau dd hudng din HS chiing minh bing cac cau hdi sau day:

HI. Hay ap dung dinh li ve bat ding thiic Cd-si cho ba sd a, b, c.
. 1 1 1
H2. Hay ap dung dinh li v6 bat dang thiic Cd-si cho ba sd
a b
H3. Hay chiing minh bat ding thiic tren.
H4. Dau bing xay ra khi nao?
Thuc hien H2
Hoat dong cua GV

Hoat dong cua HS

Cau hoi 1
Phat bieu ke't qua tirang tir he
qua tren cho trudng hgp ba sd
duang.
Cau hoi 2
Ne'u bd di dieu kien ba so
duang thi ke't qua con dung
hay khdng?

Ggi y tra 161 cau hoi 1
Neu ba sd duong thay ddi nhung co
tong khdng ddi thi tich cua chiing Idn
nhat khi ba sd do bing nhau. Neu ba
sd duong thay ddi nhung cd tich
khdng ddi thi tong ciia chiing nhd nhat
khi ba sd dd bing nhau.
Ggi y tra 161 cau hdi 2
Khdng.

14


TOM TAT BAI H O C
1.

Cac bat dang thirc cd dang a < b, a > b, a < b, a > b.

2.

Cac tinh chat ciia bat ding thiic.
Tinh chdt
Dieu kien

Noi dung
a
a>0

a < b <^ aa < ba

a<0

aa
a >0, c> 0
n nguyen
duong
fl>0

+ c
a < b va c < d ^> ac < bd
a0
<&2n + ^
a^" < b^"

a a <b ci> \fa
3.

Bat ding thiic Cd-si:
Trung binh nhan ciia hai sd khong dm nho han hoac bang
trung binh cong ciia chiing.
ab < ^ t j ^ .

Dang thicc yjab =

a +b

Va,b>0.

xay ra khi vd chi khi a = b.

Cac he qua:
He qua 1

Ne'u X, y cimg duang vd cd tong khdng ddi thi tich xy Idn
nhdt khi vd chi khi x = y.
15


He qua 2
Neu X, V Cling duang va co tich khong doi thi long x + y
nho nhdt khi vd chi khi x = v.
Bat ding thiic chira dau gia tri tuyet ddi.
Noi dung

Dieu kien

\.\-\ > 0. Ixl > X. |x| > -X
x| < a <:> -a < X < a

a>0

|x| > a <=> X < - a hoac x > a
\a\ -\b\ <\a + b\ < \a\ + \b\

M O T SO C A U H O I T R A C N G H I E M
I. Hay di6n cac dau > hoac < vao chd trdng sau day :
(a) a" + b^

2ab;

(b) b + c"

(c) a + C

2ac;

(d) a + b + c

2

2

be;
2

ab + be + ca

Tra Idi.
Cau

(a)

(b)

(c)

(d)

Di6n

>

>

>

>

2. Trong cac khang dinh sau, hay chgn khang dinh diing vdi mgi x.
(a) x^ > X

(b) X' = X

(c) 2x- > -X

(d) 2x- > X-

Trd Idi. (d).
3. Hay chgn khing dinh diing trong cac khing dinh sau :
(a) X + Ixl > 0

(c)-2x+ x <0
16

(b) X - Ixl > 0
(d)x + 2 X < 0 .


Tra lai. (a).
4. Hay dien cac dau (>, <, =) vao cac chd trdng thich hgp sau

(a) V2

V3;

(c) 72 + Vs

73 + V? ;

(b) Vs

VV

(d) 7I0

722

Tra lai.
Cau

(a)

(b)

(c)

(d)

Di6n

<

<

<

<

5. Hay di6n diing - sai vao cac cau sau :
(a) 4 > 2 ^ 4a > 2a ;

D

(b) 4 > 2 => 4a < 2a ;

D

(c) 4 > 2 ^ 4a > 2a Va > 0;

D

(d) 4 > 2 ^ 4a < 2a Va < 0.

D

Tra lai.
Cau

(a)

(b)

(c)

(d)

Di6n

S

S

D

D

6. Hay dien diing - sai vao cac cau sau day :
(a) 2006 > 2005, a > b => 2006a > 2005b

D

(b) 2006 > 2005, a < b ^ 2006a < 2005b

D

(c) 2006 > 2005, a > b => 2006 + a > 2005 + b

D

(d) 2006 > 2005, a > b =^
^

2006 2005

> --—
a
b

^
U

Tra Idi.
2.TKBGOAIS610(NC)

17


Cau

(a)

(b)

(c)

(d)

Dien

D

D

D

S

7. Hay dien dung - sai vao cac cau sau day :
(a) 5 > b ^

D

75 > 7 b

(b)5>b>0^75>7b

D

(c) a > b ^

D

7 a > \/^

D

(d) a > b ^ a" > b" n e Z
Tra Idi.
Cau

(a)

(b)

(c)

(d)

Dien

S

D

D

S

8. Cho a, b la 2 sd cung dau. Hay chgn khang dinh diing trong cac khang dinh sau:
(a) ^ . ^ ^ 2 ;
b
a
(c)

a

b

—-I- —

b

b

a

a b
(d) —+ — <0.
b a

>2

a

Tra Idi. (c).
9. Cho a > b > 0. Hay chgn ket qua diing trong cac ke't qua sau :
a
a+1
(a) - <
b
b+1
a
a+ l
(c - >
b
b+1
Tra Idi. (c).

(b) - < b
a
(d) ca ba cau dtu sai.

10. Hay chgn ket qua diing trong cac ket qua sau
(a) a -H 1 > a -K a Va e :i:

18

(b) a"^ + 1 < a'' -h a Va

G

J<;


(c) a^ + 1 > 1 V a

(d) a^ + 1 = - Va
2

Tra lai. (c).

a
1
11. Cho — < - (vdi b > 0) hay chon ket qua dung trong cac ket qua sau
b
2
^ a+1 1
(a)
< b
2
^ a+1
1
< (c)
b+2
2

Tra Idi. (c).

.u^ a + 1
1
(b)
<
b
2
a(d)
b<2

12. Hay chgn diing - sai trong cac cau sau
(a) |a| - 1 < | a - l |

(c) |a| + 1 > | a - l |

•
•

(d) |a| + 1 > a

D.

(b) |a| - 1 < |a + l|

Tra Idi.
Cau

(a)

(b)

(c)

(d)

Dien

D

S

D

D

13. Hay chgn ket luan dung trong cac ke't qua sau
(a) |x| < l<:^x= 1;

(b) |x| < 1 « - 1 < x < 1;

(c) Ixl < -1 <» -1 < X < 1;

(d) |x| < -1 c^ X = - 1 .

Trd Idi. (b).


14. Chgn bat ding thiic dung trong cac bat ding thiic sau

(a) x ' + X - 1 > 0 Vx;

(b) x^ + x + 1 > 0 Vx;

(c) - x ' + X + 1 > 0 Vx;

(d) x^ - X - 1 > 0 Vx.

Trd Idi. (b).
15. Chgn ket qua diing trong cac ket qua sau
(a)x^+l
( b ) x ^ - x + l >OVx;

(c) x^ - X + 1 = 0 Vx;

(d) x^ - X + 1 = 0 tai X nao dd.

Trd Idi. (b).
16. Chgn ket qua diing trong cac ket qua sau
1 2
(a) 0 < a < 1 thi - <
;
a
a+1
1 2
(c) 0 < a < 1 thi - =
;
a
a+1

Trd Idi. (b).

1 2
(b) 0 < a < 1 thi - >
;
a
a+1
1 1
(d) 0 < a < 1 thi - <
a
a+1

17. Chgn ket qua diing trong cac ket qua sau
(a) a+ b + c > 3\Jabc
( b ) - a - b - c < - 3¥abc

;

(c) yfa + y/b +y[c> 3^-Ja +4b+4c
(d)

\

y[a-4b+yfc>3\Jyfa+ylb+^fc

Trd Idi. (c).
18. Chgn ket qua diing trong cac ket qua sau
(a) a+ b + c > 3¥abc

;

(c) - 7 a - 7 6 - 7 c < - 3 ^ / 7 a + 7 ^ + 7 c ;
Trd Idi. (c).
20

( b ) - a - b - c < - 3¥abc

;

(d) 7 o - 7 ^ + 7 c > 3 x / 7 a + 7 ^ + 7 c


H U O N G D A N B A I TAP

SGK

Bail.
GV: Hudng dan hgc sinh Idm bdi tap nay a nhd.
Cau hdi
Cau hdi 1
Nhan xet gi vd dau ciia hai sd
a va b.
Cau hdi 2
Hay dien vao chd trd'ng sau:
1 = 1-1 = - -1. bu. . . . —.a
a
ab
ab
b
Cau hoi 3

Giai thich tai sao:
— b <—a
ab
ab

Ggi y tra 161
Ggi y tra Idi cau hdi 1
Hai so a va b ciing dau.
Ggi y tra Idi cau hdi 2

- - — b < —a--!a ab
ab
b
Ggi y tra 161 cau hdi 3
1
Vi — > 0 va b < a.
ab

Bai 2.
GV: Hudng ddn hgc sinh Idm bdi tap nay a nhd.
- On lai tinh chdt ba canh ciia tam gidc.
- Cdc tinh chdt ciia bdt ddng thiic.
Cau hdi

Ggi y tra 161
Ggi y tra Idi cau hdi 1

Cau hdi 1
Ne'u ba canh ciia tam giac la a.
a+ b+c

b, c. Tinh nira chu vi p ciia tam P "
2
giac.
Ggi y tra Idi cau hdi 2
Cau hdi 2
a + b + c-2a
b + c-a
P - ' ^
2
=
2
Hay tinh p - a, p - b, p - c.
,
a+c- b
p-b=
2
p-c=

b+a- c
2

21


Cau hdi 3
Chimg minh bai toan tren.

Goi y tra Idi cau hdi 3
Vi — > 0 va b < a.
ab

Bai 3.
GV: Hudng ddn
HS dn tap lai cdc tinh chdt ciia bdt ddng thicc.
- Mot sd ki ndng bie'n ddi bdt ddng thii'c.
Ggi y tra 161

Cau hdi
Cau hdi 1
Hay dua bat ding thire ve dang
f(x)>0.

Ggi y tra 161 cau hdi 1
a^ + b^ + c^ - ab - be - ca > 0

Ggi y tra Idi cau hdi 2
Cau hdi 2
Hay chung minh bat ding thiic Nhan ca hai ve vdi 2 ta cd.
tren.
2a^ + 2b2 + 2e^ -lab - 2bc - 2ca > 0
«> (a - b)2 + (b - c)2 + (c - a)2 > 0.
Cau hdi 3
Da'u bing xay ra khi nao?

Ggi y tra Idi cau hdi 3
a-b=b-c=c-a=0
tire la
a = b = c.

Bai 4.

Hudng din cau b)
Cau hdi

Ggi y tra 161

Cau hdi 1
Ggi y tra 161 cau hdi 1
Nhan xet ve dau cua hai ve Hai ve ciia bat ding thiic khdng am.
Ggi y tra Idi cau hdi 2
ciia bat ding thiic.
22


Cau hdi 2
Hay binh phuong hai ve va
chung minh bat ding thiie

(7a + 2 + 7a + 4) < (73" + 7a + 6)'
+ 2 + a + 4 + 27(a + 2)(a + 4)

hay la

< a + a + 6 + 27a(a + 6)
nen

^(a + 2)(a + 4) < 7a(a + 6).

Do dd (a + 2)(a + 4) < a(a + 6) hay la
a^ + 6a + 8 < a^ + 6a nen 8 < 0.
Dd la dieu vd li.

Vay
7a + 2 + 7a + 4 > 7a + 7a + 6 (a > 0).
Cau a) la dang dac biet cua cau b) vdi a = 2002.
Bai 5.
Hudng ddn hgc sinh Idm bdi tap nay
- On lai dinh li bdt ddng thiic Cd-si cho hai so khdng dm.
Cau hdi

Ggi y tra Idi

Ggi y tra Idi cau hdi 1
Cau hdi 1
Nhan xet ve dau ciia hai so Hai so nay cd dau duang.
1 . 1
— va —.
a
b
Cau hdi 2
Ggi y tra Idi cau hdi 2
Ap dung bat ding thiic Cd-si
chi hai so — va —
a
b
Cau hdi 3
Hay so sanh —j= va
yjab
a+b

l a CO — + —>

a

b

.—

^ab

Ggi y tra Idi cau hdi 3
/—
2
4
Ta cd a + b> 2\Jab nen —j= >
7a/)
a+b
23