TAO Mm THU CHO HOG SINH THONG QOA CAC DAI TOAN v i BAT DAriQ THUC (TOAn 10) I Trong giang day, phat huy tinh tich ci/c cua HS Iadieu quan trpng nhatcua noi dung doi m6i phuang phap De lam datfcdieu na[.]
Trang 1TAO Mm THU CHO HOG SINH THONG QOA CAC DAI TOAN
v i BAT DAriQ THUC (TOAn 10)
I.Trong giang day, phat huy tinh tich ci/c cua HS
Iadieu quan trpng nhatcua noi dung doi m6i phuang
phap De lam datfcdieu nay, moi giao vien (GV) can
dau tir thdi gian,lu6n tim toi vaphathien nhi/ng vande
m6i, tur do hiding HS den vdi "the gidi rang md' cua
loan hpc, lihol day long dam me toan hpc dcac em
Trong qua Irinh day hpc, chung toi lhay cd nhiing bai
tap nhin qua tirdng chi)ng nhir rat don gian, nhung
neu chju kho tim hieu Ihi se kham pha dirpc nhieu
dieu thu vjtdnhirng bai toan dd Bai viet nay chung toi
trao doi cung dong nghiep va dpe gia mpt so'bai toan
ve bat ding IhiJc (BBT) trong chuang Irinh toan hpc
Idp 10 BBT la mpt phan quan trpng trong chuong
Irinh Toan phd'thdng, nd ludn dUdc decap den Irong
cac ki thi HS gidiva trong cac ki thituyen sinhdai hpc
Tuy nhien, day la mpt mang khd vi vay da so' HS
thudng bd qua c&u nay GV phai iam sao de HS tie'p
thu dupc kien thi^c vephan nay mpt each chu dpng,tl/
dd cac em hiing thu hon trong hpc Toan Be nang
cao hung ttiu hpc tap men Toancho HS, Ihdng thudng
sau khlHS lam xpng mpt bai toan nao dd, GV can dat
cho HS cac cSu hdi sau:-Lieu bai toan nay cd trudng
hpp rieng nao khdng?; - Cdthetd'ng quat baitoan nay
dupc khdng?; - Cd the ap dijng bai toan nay de giai
cac bai toan khac dudc khdng?
Chinh nhi?ng cau hoi nay lam cac em phai suy
nghlva Cling vdi suhudng d i n cua GV cac em cdthe
tim ra duoc cau tra idi T i i dd, cac em yeu Ihich hpc
mdn r o i / r h o n Vidu, baitoan: Chiing minh ring neu
a, b, c i a b a s 6 ' d u a n g l h i C ' ' + ' + c ) ( - + ^ + - | s ^ 9
Khi nao xayra d i n g thiic?
Viec chiing minh baitoan nay kha don gian, sau
khi MS chiing minh xong, GV nen dat 3 cau hoi:
• BDT Iren tuong duong BDT nao?; - Neu xet vdi hai
sfiduong thi ta cd BDT nao?; - Lieu bai loan cd diing
cho nsd'duong khdng?
Tilcac cau hdi Iren, da giiip cac em lim ra dupc
cac BDT cd nhieu ting dung trong viec giai cac bai
toan trong cac ki thi tuyen sinh dai hpc trong nhiing
nam gan day:
ThS LE Q U A N G H A O '
- Chiing minh ring ne'u a, b, c ia ba sd duong thi
- Chiing minh ring nd'u a, i> ia hai sd duong Ihi ( " + ' ) : ^ + ^ 1 ^ 4, Khi nao xay ra dang thiic? I M
- Chiing minh ring ne'u a, b la hai sd duong thi
- + T ^ — 7 Khi nao xay ra dang thdc?
a b a+b ' ^
• Chiing ring neu a,, a^ a„ iacac sdduong thi
(o,+a,+ +o.) — + — + + — > n ' K h i n a o xayra
d i n g thiic?
2 Tao hixng thu cho HS qua khai thac "bai toan gd'c"
1) Bai toan goc Bang nhiing kinh nghiem thi/c
tien giang day, GV cd the chpn cac bai toan trong SGK cd tinh khai quat cao (tuong dd'i) iam "bai toan gd'c" Ve BDT (Dai sd'10), chung tdi chpn "bai tpan gd'c":
1) Chiing minh ring neu a ^ 0 va 6 > 0 ttii
a ' + i ' > a i ( a + i ) ( b a i 6 , t r 110-OaisolOnangcao) Chung minh:BDTtuong duong (o+*)(a^-2ii*+(>^)
o (o+6)(a-6)^ >o BDT dung, da'u "="xay ra khi va chi khia = d
2) Chiing minh rang vdi hai sd thuc a , dtuyytacd:
a*+b*>a^b+ab^ (dai 7b, tr 110 - Dai so 10
nang cao)
Chiing minh baitoan nay tuong ti/bai 6 Sau khi chiing minh xong hai bai loan neu tren, GV can gpi y cho HS khai thac t?o hung thu qua giai bai loan 6 va bai 7b nhusau:
2) Tao hung thu hgc toan qua "bai loan goc"
- Hudng khaithae dan gian: N eu chii y, bai toan 6
va bai 7b cd the tong quat nhu sau:
* Tnriing THPT Chuyen Huynh Man Dat tinh Kien Giang
(ki2-8/2014) Tap chi Gide due so 340 51
Trang 2+ Bai6.1 Chung mmh r§ng neu a > 0 va 6 > 0 thi:
^2n+i ^^2n+i > ^2n£,^^^2n {yoj n nguyen, di/cfng}
Thatvay, BDTtaang duang ( a ^ " - 6 ^ " ) ( a - f e ) > 0 ,
BDT dung, dau "=" xay ra khi va chi khi a = b
+ Bai 7b.1 ChiJng minh rang vo'i hai so thi/c a, b
tuy y, ta c6: a^" + fp-" > a^"-^b+a6^""' (vert n nguyen,
du"cfng)
Chung minh tUtfng tubai6.1
Tuket qua tren, luu y HS khi giai mgt bai toan nao
do ta CO the chu y den cac tnj'dng hffp rieng (neu co)
hoac ta co the tong quat hoa bai toan
- Hudng khai thac muc do ning cao: Neu v6i
a>0,b>0, c>0{Xilba\^.^){a CO ^+—>a+b,
— + ^ > c + 6 , —+—>a+c Cong cac BDT tren
C b c a • ^
ta CO bat toan 6.2
+ Bai 6.2 Cho a, b, c\a ba soducfng Chung minh
rang:_
>2(a + b + c) b^ +a^
Bai toan 6,2 tuong dUdng
a\]- + -) + b\- + -) + c\]- + -)>2{a + b + c) «
DC a c b a
a \ ^ ) + b\^^) + c\^)>2{a + b + c).Qala+b
' be
, 1 - ,,1-A, i , l - c ,
+ c = 1, ta co: a\^)+b\^-^)+cH^)>2 «
be ac ab
o'(i-a)+6^(i-z>)+c^{i-c)>2i76(?.Dendaytac6bai
toan 6.3
+ Bai 6.3 Cho a, b, c la cac so duong thoa man
a + d + c = 1 C h i j n g m i n h : f l ' + 6 ' + c ' > a ' + 6'+c' + 2aAc
Neutadata= 3 / j , b = ^ , c = i/^.Talaic6them
baitoan 6.4
+ Bai 6.4 Cho x,y, z laba so duong thoa man ^
Neu ta coi a, b, c la do dai ba canh cua mot tam
giae thi tu'6.2 kel hop djnh li sin ta c6 bai toan:
•f Bai 8.5 Cho tam giae ABC Chung minh r^ng:
sin^.^ + sin^ B sin^fi + sin^C s'm^C + sin^ A
sinC sin A sinB
2{sinA+sinB-h sinC)
Den day, neu ta ap dung cong friuc sm'« = u cos' a
ta duoc bai toan 6.6
+ Bai 6.6 Cho tam giae ABC Chung minh rang:
£2{sin/* + mC)^
Neu xuat phat tur bai 7b.1 vert /? = / ta co BDT quen
thupc o^ + 6' > 2ab vdi moi a,/jthupc R; vert /7=2ta co bai toan 7.b.2: a'^+b'* >a^b-i-ab^ (1) vcri mpi a, b thupc R;neu ta thay boi gl vdi a ^0, b ^0 khi do (1) tuong duong ^ j ^ ^ ab xuang \\sXac6-T^^ be,
J "^ " a+b • b +c^
V ^ s ^c Cong cac BDT Iren ta co bai toan7b,2
c +a + Bai 7b.2 Cho a, b, e la cac so thue 5^0, Chung
minh r^ng:
a''+b^ V+c"*
a^+b^ 6^+c^ c' +0' yd\n = 3 Xa co BDT a* +b^ > a^b+ab^ (2) vdi mpi
a, b thupc R, Tuong l u nhu tren la co bai toan 7b.3
+ Bai7b.3.Choa, 6, clacaesolhLrc^O.Chiing minh r&ng:
a'' + b'' b''+c'' a''+b'* b'' + c'' lis bai 7b.2 va bai 7b.3, tong quat thanh bai toan
7b.4
+ Bai 7.4 Cho a, b,c\a eae so thi/e ?iO Chung
minh r i n g :
j-tab + bc+ca
rtab + bc + ca
Jh'*l.L2^*2 L2«*1 , „ 2 B + 2 „2«t2 , „2iit2
• +—-—T:—>flO+f>c + cfl
a" +b b " +c c"' +a"'
(vdi/T nguyen,duong)
Tai l i | u tham khSo
1 NguySn Huy Doan (chu bien) Dai stf 10 (nang cao)
NXB Gido due E 2006
2 Tr^n Van H90 (tdng chu bifin) D^i stf 10 NXB
Gidodfic, H.2001
SUMMARY
In teaching, improving students activeness is the most important thing when we want to renovate our methods To make it work, each of us, teachers, must spend a lot of time studying and discovering new things As a result, we can lead our students to the new open horizon, making them more interested in our subject Through my teaching period 1 notice that some exercises seem to be simple However when we try our best to solve them, we can get many interesting and surprising from them
52 Tap chi Giae due so 340
(kl2-8/2014)