) Rat nhi6u bzur dI nhdm khi dua ra d6nh giii: Do a < 4 ndn a(3 2b < 4(3 2b (l
cos2 3a=cos23l :-b | =cos2 3b=mi (3 )
(3 )
/- \
cos2 Jr=cos2:{ 1+b I : cosl 3b=ttt'
\3 )
vi (4cos3 b-3cosb)2 =cos2 b(4cos2 b-3)2 :nt
hay phuong trinh 1613 - 24i') + 9t - m = O,
hay phuong trinh 1613 - 24i') + 9t - m = O,
Suy ra phuong trinh
mu3 - 9u2 + 24u - 16 = 0 c6 cdc nghiOm
111lrt = lrt = -, UZ=---7---\ . rrf =---71 \. crx' b "n*'I ---[: -) I -a ] .or'{ [3 l *r l )
Khi d6 P = u? +u]+al. Su dung ho thric
Vidte r,) ding thric ul + u] +ul = (n1 * Lt2*L\)3 -3(r1 +u2)Qr2 *lt1 )(rtj *a| ), ta thu duoc 1s\3 (s \re \r9 \ P=l- I -31--rrr ll--ut ll --rrr I \m) \rt )\rrt )\tn ) lt; q
hay P=P(x)=ir3-8x2+I;, .u=129.
(Do0<m<l).
NhAn x6t rang hhm so ndy c6
16,
P'(.tl=3.r2-16r+I>0. Vx > 9. nen P(.r)
cl6ng bien trong [9, im).
Sr"ry ra minP = P(9) = 129, dat duoc khi ri = 1
hay cos23b = 1 e sin3b =0 e b = +.Doci6
-)
a=(A -l)1, c=(t+l )L,k e Z. fr
I
JJ
{Nnan x6t. Day li mot bari to6n vd bien ddi v}r tinh
tofn luong gidc khd phric tap. Da so cic ban ddu giiri tudng tu nhu ciich da trinh bly o tren. Mot so ban'cdn cho ciich chrlng minh truc ti€p bat dang thric P > 129 dua trtn nhan x6t cos'30 < 1" M6t sd khdc str dung cong th(rc khai tridn
(n \ (n ) 3ianb-tanrb
ranlb = tanlsl -(: :-b I ltanbl l-a I -+b l= l-3tanzb-
v) cons " ttrtc -f coszb = l+tanzb dd dua bii toiin vd x6tdang phuong trinh bAc ba theo t = tanb vi tinh P. Cic dang phuong trinh bAc ba theo t = tanb vi tinh P. Cic
bni giAi ddu cho d6p s6 dring.
NGUYEN VAN MAU
*gai TDl367. Cho hdm sa f@ xdc dinh tran
itoqn l0 ; 17 thod mdn ba ttnh chdt sau:
i)(l):1;
)rx) =
i( r(;).r(*).r(+)
vdi mpi xe [0; l];
iii) Vdi mgi sii thwc dtrong e nho tity y, lu6n c6
m\t sd duong 6, (6, phu thuQc vdo e) sao
cho; Vdi mqi x, y e [0 ; 1] md lx - yl < 6, thi
l/U)-fu)l<e.
Chr?ng minh riing /(x) : 1 vdi mqi xe [0 ; l1 .
Loi gidi. Theo gid thiet vdi moi sd thuc duong e, luon tdn tai m6t so thuc duong d sao cho neu l.r -.vl < dthi I f @)- f 0)l< e .
X6tx, y € [0 ; i] tLy f m] l-r*yl < 3dta c6
I f(x\- 1 Q'r :jll( I (+),(+))l 'ifl4+)-,(+)l " do rtx) = 1i / [,{]:), f6 r=-L'l r[r: j 3ft'[ 3 ) - 3; \ 3 ) VA ..t,
,r+i y+il lr-vi r t_t / / -\
-\y la
JJI J
r,6i moi i = A, 1,2.
L4p lai tudng tu ta suy ra !/(-r) - f(y)l < e v6i
moi -r, y e [0 ; 1], lx - -]'l < 3''.d vi vdi moi
m e N. Nhuv4y i/(r) -f(y) | < e v6i moi
r, ) € t0 ; li. Khing dinh tr6n dring v6i moi
s> 0" Do d6/(") =f!),vdi moi x, y e [0 , 1]. B<ri
v0y.flr) =J'(1) = l, vdi moi.r e [0; 1]. KhAng rtinh c[ra bii todn dd cluoc chrlng minh . D
{t*ne, x6t. Day l} bii torin giai tich dang co ban,
khdng quii kh5. Ngoli crlch girii trdn cdn c6 ciich girlri
khiic: dirng su kien hlm so f(-r) xric dinh liOn tuc fren TO6N HOC