m + 2 4 > 0 o m > 6 . ,
Dicu kien du: Gia suT m > 6 ' ,. . Ta cd x^ - 2x + m = (X - 1)^ + m - 1 > 5 Vx G [-4; 6]. Ta cd x^ - 2x + m = (X - 1)^ + m - 1 > 5 Vx G [-4; 6]. Theo bat ding thufc Cdsi, vdi moi x e [-4; 6] tW
tf4 Q SR
..(•1.'
Tiirdd suy ra khi m > 6 thi V(x + 4)(6-x) <x^ -2x + m Vx G [-4; 6] ^hu' vay m > 6 la cac gia tri can tim cua tham so m. ^hu' vay m > 6 la cac gia tri can tim cua tham so m.
'^o rang each giai nay cung v6 ciing ddn gian va sang sua!
%h 4: (PhircJng phap do thi)
y = 7(x + 4)(6-x), thi ta cd y > 0 va cd: ' ? ^
ChuySn dg BDHSG Toan gia tr| lOn nha't gia tr| nh6 nhat - Phan Huy Khii
y > 0
- x ^ + 2 x + 24 = y^
y >()
( x - l ) ^ + y ^ =25
Vi the do thi cua y = 7(x + 4 ) ( 6 - x ) la nuTa diTcJng Iron (phia tren true hoanh)
tarn tai diem 1(1; 0) va ban kinh R = 5. : y'
y = x^ - 2x + n la parabol vdi moi m va deu nhan x = 1 la true do'i xiJng.
Bai toan c6 dang: Tim m de
parabol la luon luon nkm tren
nuTa dU'cJng tron.
Xet parabol ticp xuc \6i nua
di/dng tron tren tai M ( l ; 5). Dieu do xay ra khi va ehi khi m - 1 = 5 <=> m = 6.
Vay cac parabol can tim la tjnh ticn parabol ay len tren, ttfc 1 in > 6. Ta thu lai ket qua tren.
Cdch 5: (Sit dung tam thtfc bac haD'^'^fV^-it! m.'^v ' ^ ' - I f e .
Bai toan da eho CO dang: »i ' K r K ' Tim m de bat phiTcJng trinh f(t) = t^ + t - 24 - m < 0 (*)
4 u n g v d i m o i 0 < t < 5 .
Ta ehi xet khi A > 0 (vi neu A < 0 thi t^ + t - 24 - m > 0 Vm trif ra eo the chi 1
b^ng 0 tai mot diem t = - - ) .
Luc do (*) o t| < I < t2. Vay ta can eo [t,, h] c [0; 5]
< : : > t i< 0 < 5 < t 2 O f ( 0 ) < 0 - 2 4 - m < 0
6 - m < 0 o m > 6. Ta thu lai ket qua tren.
Chung toi da trinh bay 5 each giai khac nhau (trong do eo hai each diing (adsbygoogle = window.adsbygoogle || []).push({});
gia tri Idn nhat va nho nha't cua ham so) bai toan tren. Ban c6 nhan xet
Bai 2. Tim m de bat phiTdng trinh - 4 v ' ( 4 - x ) ( 2 + x) < x^ - 2x + m - 18 dung
moi X e [-2; 4]
HUdng Mil giai
Viet lai bat phu'dng trinh da eho du'di dang
( x ^ - 2 x - 8)+ 4V-x^+ 2 X + 8 - 1 0 > - m (1)
Cty TNHH MTV D W H Khang Vi$t
Pat I = V-x^ +2x + 8 . Xet g(x) = -x^ + 2x + 8 vdi - 2 < x < 4.
Ta CO g'(x) = -2x + 2, va c6 bang bien thien sau: 7
X - 2 1 4 g'(x) 1 + 0 g(x) 1 ^ ^ ^ ^ 1 . Taco max g(x) = g(l) = 3; ' f - 2 < x < 4
min g(x) =min{g(-2);g(4)} =min{0;0} = 0 . •
- 2 < x < 4
Vay bie'n mdii t eo mien xae djnh 0 < t < 3. •?* * ' > ~ ' '' ^ 1
Luc nay (1) CO dang - t " + 4t - 10 > - m
<=> f(t) = t ' - 4 l + 1 0 < m . (2) Bai toan da eho trd thanh:
Tim m de (2) dung vdi moi t e [0; 3|.
Dieu d6 xay ra khi va chi khi max 1(1) < m . (3)
()<i<.^
Ta CO f'(t) = 2t - 4 va eo bang bien thien sau:
I 0 2 3
f'(t) I'd) 1 0 + 1
1 ^ ^ ^ ^
Vay max f(t) = max{f(0); f(3)} = max{ 10; 7} = 10. (4) TiY (3) (4) suy ra m > 10 la cac cac gia tri can tim cua tham so m
l^hdii xet: Khac vdi bai 1, bai nay ehi c6 each giai nay la hdp li nhat. Cac ban thu" 11 giai xem VI sao? .L . i.Vfei
3. Cho bat phirwng trinh • •
3cos'*-5eos^x-36sin^x-15cosx + 36 + 24m-12m^ > 0
Tim m de bat phiWng trinh dung vdi moi x e R. ] , "
HUdiig dan giai
Dtfa bat phiTdng trinh da cho ve dang
Scos'x - 5(4eos'x - 3eosx) - 36(1 - eos^x) - 15eosx + 36 + 24m - 12m^ > 0 <t> 3cos''x - 20cos^x + 36cos\ 12m^ - 24m. (1) (adsbygoogle = window.adsbygoogle || []).push({});
Dat cosx = t (-1 < t < 1), khi do bai toan da eho trcf thanh
Tim m de bat phiTdng trinh f(t) = 31" - 20t' + 36t^ > 0 12m^ - 24m dung vdi m p i - 1 < t < 1.
2?,9
Chuygn dS BDHSG Toan g\A trj I6n nh^t vA gia tr| nh6 nh^t - Phan Huy Khii