Tam thức bậc hai và một số ứng dụng

M CL C I: M T S D TAM TH C B C HAI mc c hai 1.1.1 Nghi m c c hai - u c a tam th c b c hai n v d u c a tam th c b c hai o v d u c a tam th c b c hai 10 m c a tam th c b c hai v i m t s n lu n c hai 11 NG D O V D U C A TAM TH C ,B B C GI CH A THAM S 15 2.1 H 2.1.1 H nh ad u i 15 n h p 15 ad i 16 2.2 D u c a tam th c b t mi i bi n lu n b 18 c cao 23 23 c cao 26 31 31 33 2.5 M t s 35 2.6 M t s sai l m c a h c sinh s d o v d u c a tam th c b c hai 37 C NG D O V D U C A TAM TH C B C KH PH THU C THAM S 39 c ng bi 39 ngh ch bi t mi n 40 b c 3: 41 c: 43 3.3 C c tr th c 44 tham s u ki nh t th 3.5 S l n nh nh c 46 th b c ba v i m ng th ng 48 mc ng th ng v b cb c a hypebol 52 mc mc : TH C B ng th ng v ng th ng v th c 52 a hypebol 53 NG D O V D U C A TAM C CH NG MINH B T NG TH C 55 4.1 ng d o v d u c a tam th c b c hai vi c ch ng minh b ng th c 55 4.2 ng d o v d u c a tam th c b c hai vi c gi i c 57 K T LU N 59 DANH M U THAM KH O Error! Bookmark not defined Thang Long University Libraty M Trong c gi ng d c uk v is lu d U ng ph i s l p 10 vi t v tam th c b c hai C ng d ng nhi u n gi , kh ng ph c gi ng d ab ct c b N i dung lu c t t qu c a n lu n u ng d y ts ng n sau: I M t s d h th i m tam th c b c hai ng n g n v m c a tam th c b c hai v II ng d o v d u c a tam th c b vi c gi ts ng d ng tr c ti a d c cao, gi m ts d o c a tam th c b i ih i, gi nh p , gi i , d u c a tam th c b t mi i bi n lu n b , m t s sai l m c a h c sinh s o v d u c a tam th c b c hai III c kh ts ngh ch bi tham s u ki c ng d n o v d u c a tam th c b c hai T d o c a tam th c b c hai kh th ng bi n, t mi n, c c tr d th c ,x l n nh nh nh t th IV ng d ng ch ng minh b ng th nh ng ng d vi c ch ng minh b ng th s Lu ng d ng th i, ih M n lu nc nh o v tam th c b c c o v d u c a tam th c b c hai m ts c b ot ih i n Cu xin ng d n c a Th y a TS iv is xin c ih u ki n thu n l tc g n, i thi mong s c gi Thang Long University Libraty C mc c hai 1.1.1 b c hai f(x) = ax2 + bx + c = (a, b, c b x a x2 f( x) b x 2a b x 2a 0) c a b2 4a 2 b2 c a 4ac 4a = b2 t *N u m =0t *N u *N u b 2a x nh (1) hai nghi y r ng n bi t x1, b 2a cb t N ' ch n, b = 2b b ' ac, ' b' x1,2 ' a v i ' 1.1 Gi n lu n theo tham s (m-1)x - 2(m-3)x + m - = (1.1) L i gi i a) N u m = (1.1) tr nghi x = b) N u m m ' N u ' c nh t 4x = c c hai v i (m 1) 4(m 2) m >2 (m 3) 4(m 2) m ' N u 4(m 2) m m x1,2 ' N u t: x1 m m 4(m 2) m m 2 m ; x2 m m m 2 m m K t lu n: m nh t x = m x1, t x1 m 2 m ; x2 m m 2 m m 1.2 Ch ng minh r (x + 1) (x + 3) + m ( x + 2)( x + 4) = (1.2) m th c m L i gi i 1.2) x2 x mx2 6mx 8m (m 1) x2 2(3m 2) x 8m m = -1 m x= m *) N u m + 2x *) N u m + 1= ' = (3m + 2)2 - (m + 1) (8m + 3) = m2 + m + 1 = m V 5=0 m 1 t m m m Thang Long University Libraty c hai ax2 bx c (a N S x1 P x1.x2 b a x2 x1 , x2 m 0) c a * Nh +) x1 x2 x1 +) x2 x1 x2 x1 x1 x2 x2 0 x1 x2 +) x1 x1 x2 x2 c l i, n u hai s th X SX P (v i S P ) mc 1.3 G i x1, x2 mc 2x2 + 5x - =0 (1.3) 1 tl y1 , y2 x1 x2 L i gi i Do x1 , x2 mc nh (1 x1 S y1 P y1 y2 x2 x 1 V y y1 , y2 Hay X x2 2 x2 1 x1 x2 1 x x1.x2 y2 x x x2 ; x2 -et ta x1.x2 x1 x2 2 X2 mc X X n l p 1.4 x2 (2m 1) x m2 (1.4) m x1 , x2 cho x1 2x2 L i gi i m (2m 1)2 4(m2 1) 0 4m m Theo h th c Vix1 x2 x1.x2 2m m2 T x2 2m 2 2m (*) x1.x2 m2 (**) 2(2m 1) 2m , x1 c: m2 m2 8m K th 3x2 Do x1 2x2 u ki n m 2(2m 1) 9m2 8m2 8m 9m2 m m c m = Khi gi i m u ki n c a tham s tho uv u ki mc 1.2 u c a tam th c b c hai a tham s 1.2.1 Cho tam th c b c hai f ( x) ax2 bx c, a N u uv ih s av im ix N u u v i h s a, x0 N u m x1 , x2 ( x1 x2 ) u v i a v i m i x x1; x2 b2 4ac 0, b t i x0 2a b , f ( x0 ) 2a u v i a v i m i x ( x1; x2 ) y: a f ( ) x1 ng h ( ) a f ( ) x2 Thang Long University Libraty 5x2 1.5 Gi i b L i gi i Tam th c v x1 2; x2 ' 64 t u v i h s a = -5 < nghi m c a b ( 1.6 V i nh nghi m (m 1) x2 2mx (m 3) L i gi i * TH1: m = -1 b nghi m x * TH2: m v 2m2 a m a + S S 7 m2 2m + + a + _ + suy b x2 ; m +) Khi m 1 ; x1 (1.6) + +) Khi m sau m m m -1 - ) (2; h cb 2m m ng sau: ) ; p h p 2x + m ho ng h m x1 x2 (- ; x1) (x2; + ) H qu Cho tam th c b c hai f(x) = ax2+bx+c (a t s th c , a.f( kh m x1, x2 th 54 Thang Long University Libraty 4.1 ng d o v d u c a tam th c b c hai vi c ch ng minh b t ng th c Vi c ch ng minh b nhi b ng th c h c sinh ph m , ta ch ng th c d c pt ng minh o v d u c a tam th c b c hai n th c b ch cr t o v d u c a tam gi i quy t 4.1 a1, a2, , an < < 1; i = 1, , n ch ng 2 minh (1+ a1 + a2 + + an)2 > 4( a12 a a n ) L i gi i 2 f(x) = x2 - 1(1+ a1 + a2 + + an) x + ( a12 a a n ) T (a12 f (1) ak ak ak a1 ) (a ak a ) (a n f (1) V a n ) Do < ak < o c a tam th b (1 a1 a a n ) k 1, , n 4(a12 > hay 2 a a n ) c ch ng minh 4.2 Cho s th c a, b, c ch ng minh r ng n u t n t i s th c m th a am2 c bm b2 4ac L i gi i f ( x) +) N +) N u a ax2 bx c 4ac > t gi thi t am2 c bm (am2 c bm)(am2 c bm) (1) 55 ( am2 c)2 (bm)2 +) N u m = t gi thi t +) N u m m + bm + c bm + c T -m) ax2 t (1) m (t = b2 4ac o v d u c a tam th c u ph i ch ng minh 4.3 Cho a > Ch ng minh r ng d -m f(-m) = am2 b c hai) b2 4ac c=0 c hai v a a a 4a ts t L i gi i t xn = a xn a a a 1d a a n => xn c hai v a a > xn-1 xn a xn xn d u c a tam th c b c hai v ch c f(t) = t2 ts t ng quy n p) xn a t ov n < xn < t2 v i t1, t2 4a V y xn 1 m c a f(t) v i 4a 4.4 Cho x t2 ts t y ch ng minh: x2 3xy L i gi i t f(x) = x2 + +) N u y +) N u y a.f ( ) = x x 6y = 9y2 < => f(x) > x x R >0 a.f ( ) = 6y 5 56 Thang Long University Libraty s 3y 2 2 y x2 (x1, x2 x1 x V y v i -1 4.2 3y 2 ng d m c a f(x)) u ph i ch ng minh o v d u c a tam th c b c hai vi c gi i c V i ng d ng t c th sau: 4.5 nh b ng G a tam t AC t i N ch ng minh: S AMN A L i gi i x y M S AMN S S AMO AON xy sin 600 O AO sin300.( x y) 2 AH N 3 B H (1) S xy t xy = t S AMN max S AMN AMN xy t max t (2) T ( mc X) = X2 3tX + t = (3) l n nh t c nghi m th u ki n < X1 < X2 < 57 C o v d u c a tam th c b t 2t a f (0) a f (1) 0 s T S AMN 9t 0 4t 3t l n nh t t = t t 2 (ho c x = , y 1) S t t = AMN t c V t lu n sau: M = B, N = N1 (ho c M = M1; N = C) SMax = N1 1, mc Smin = Hi n nh i BC < S AMN < Smax hay S AMN 58 Thang Long University Libraty c 59 , ng luy t p 1, t p NXB GD 1996 Nguy u, Nguy n Ti n Quang m th c b c hai ng ph Tr u Nguy n Danh Phan, i s 10 NXB GD 1999 Nguy n Vi t Di n, loogarit n l c gi p 10 NXB GD 2000 p 11 NXB GD 2000 Nguy n Tr ng Nh t, Nguy ch n nh Nguy ih ng, Tuy n ng NXB GD 2001 c T n, ts ng d ng NXB GD 2005 10 11 , NXB GD 2005 , ts NXB GD 2005 12 p NXB GD 2006 60 Thang Long University Libraty