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CC TR CA HM S(Chng trnh ph thng)
Nguyn Li GV THPT Chuyn Lng Vn Chnh 1.Cc tr ca hm s bc ba y = ax3 +bx2 + cx + d ( a 0 ). Gi y= 3ax2 + 2bx + c ; v y ' = b 2 3ac . + hm s c cc tr (hai cc tr) y ' > 0 ( p/t y = 0 c hai nghim phn bit). +Hai cc tr C,CT i xng nhau qua im un. +Chia a thc y cho ,hm s vit li y = (mx + n).y + px + q. Gi s (x1,y1);(x2,y2) l hai cc tr nn y(x1) = y(x2) = 0 hai to cc tr tho mn phng trnh y = px + q, ,l phng trnh ng thng i qua hai C,CT. + hai gi tr C,CT tri du nhau y ' > 0 b 2 3ac > 0 ymax . ymin < 0 ( px1 + q )( px2 + q ) < 0
( phng trnh y = ax3 +bx2 + cx + d = 0 c ba nghim phn bit ). 2.Cc tr ca hm s hu t dng y =ax 2 + bx + c (a, a' 0) a' x + b' aa ' x 2 + 2ab ' x + bb ' a ' c h( x ) = ; v h . Gi y ' = 2 (a ' x + b ') (a ' x + b ')2 h > 0 + hm s c cc tr (hai cc tr) b' h( a' ) 0 b' ). a'
( p/t y = 0 c hai nghim phn bit x +Hm s c dng y =
u u' v v ' u u u' 2ax + b y' = y' = 0 = = 2 v v v' a' v Gi s (x1,y1);(x2,y2) l hai cc tr nn y(x1) = y(x2) = 0 hai to cc tr tho mn 2 ax + b , l phng trnh ng thng i qua hai C,CT. phng trnh y = a' ax 2 + bx + c = 0 v nghim. + hai gi tr C,CT tri du nhau phng trnh y = a' x + b' 3.Cc tr ca hm trng phng :y = ax4 + bx2 + c ( a 0 ).
+Hai cc tr C,CT i xng nhau qua giao im hai tim cn.
Gi y = 4ax3 + 2bx = 2x(2ax2 + b); v tch a.b Hm s ch c mt cc tr a.b 0 ( p/t y = 0 ch c mt nghim). Hm s c ba cc tr a.b < 0 ) ( p/t y = 0 c ba nghim phn bit). +Ba cc tr to thnh mt tam gic cn. Ch : nhn bit ti im x0 l honh ca C hay CT,ta c hai du hiu: 1. Lp bn bin thin (Xem sch). 2. + x0 l im cc i y' ( x 0 ) = 0 y' ' ( x 0 ) < 0
+ x0 l im cc tiu
y' ( x 0 ) = 0 y' ' ( x 0 ) > 0
BI TP T LUYN . Bi 1. Cho ng cong (Cm) c phng trnh y = x3 3(m+1)x2 + 6(m +1)x + 1. 1. nh tham s m hm s c hai im cc tr dng.
2. nh tham s m hm s nhn x = 3 + 3 l im cc tiu. Bi 2. nh tham s m ng cong y = x4 2mx2 + m 1 c ba im cc tr to thnh mt tam gic u. Bi 3.Tm tham s m hm s y =x 2 (2m + 5) x + m + 3 c cc tr ti im x +11 c cc tr v khong cch t im cc tiu ca ca x 1
x >1.Hy xc nh l im C hay CT. Bi 4.Tm m hm hm s y = mx +
th hm s n tim cn xin ca th bng
2
.
2 x 2 + (m 2) x . x 1 1 2 Cu 6 .Tm tp hp trung im ca hai cc tr ca hm s y = x 3 mx 2 x + m + . 3 3 2 x + mx + 1 c hai im cc tr M,N . Cu 7. Chng minh vi mi m hm s y = x+m
Bi 5. Tm tp hp im cc tiu ca hm s. y =
nh m MN nh nht.x 2 (5m 2) x + 2m + 1 c cc tr v khong cch gia hai im x 1 cc i ,cc tiu nh hn 2 5 . x 2 + (3m + 2) x + 2m 1 Bi 9.Tm m d hm s y = c cc tr v ng thng i qua hai im x 1
Cu 8.Tm m hm s y =
cc tr to vi hai trc ta mt tam gic c din tch bng 2. Bi 10.nh tham s m hm s y = x 3 3(m + 1) x 2 + m + 2 c hai gi tr cc tr tri du v ng thng i qua hai cc tr i qua im M(-1;4).mx 2 + 3mx + 2m + 1 Bi 11. nh tham s m hm s y = c hai cc tr nm v hai pha vi x 1
trc Ox.x 2 (m + 3) x + 3m + 1 Bi 12.nh tham s m hai gi tr C,CT hm s y = u m. x 1 x 2 + 2kx 5 c cc i ,cc tiu v cc Bi 13. Vi gi tr no ca tham s k th hm s y = x 1
im cc i cc tiu nm v hai pha ng thng (d) 2x y = 0 . Bi 14.Tm tham s m hm s y =x 2 (3m + 1) x + 4m c hai cc tr i xng nhau qua 2x 1
ng thng (d) x+y+1 = 0. Bi 15. nh tham s m ng cong y = x3 3x2 mx + 2 c C,CT cch u ng thng (d) = x 1 . Bi 16. nh m hm s y =2 x 2 3x + m c hai cc tr tha mn y cd y ct > 8 . xm x 2 + (m + 2) x + 3m + 2 c hai cc tr khi chng minh rng Bi 17.Tm m th hm s y = x +1 1 2 y 2 cd + y ct > . 2
Bi 18. nh m y= (x 1 )(x2 4mx -3m + 1) c hai gi tr cc tr tri du nhau.