Tổng hợp tài liệu toán lớp 10 phần (9)

Khoỏ h c Toỏn 10 - Th y L u Huy Th BI B T ng Chuyờn 04 B t ng th c v b t ph ng trỡnh NG TH C V CH NG MINH B T NG TH C (PH N 1) P N BI T P T LUY N Giỏo viờn: L U HUY TH NG Cỏc bi t p ti li u ny c biờn so n kốm theo bi gi ng Bi B t ng th c v ch ng minh b t ng th c (Ph n 1) thu c khúa h c Toỏn 10 Th y L u Huy Th ng t i website Hocmai.vn giỳp cỏc b n ki m tra, c ng c l i cỏc ki n th c c giỏo viờn truy n t bi gi ng Bi B t ng th c v ch ng minh b t ng th c (Ph n 1) s d ng hi u qu , b n c n h c tr c bi gi ng sau ú lm y cỏc bi t p ti li u ny (Ti li u dựng chung cho P1+P2+P3) Baứi Cho a, b, c, d, e R Ch ng minh cỏc b t ng th c sau: a) a b c ab bc ca b) a b ab a b c) a b2 c2 2(a b c) d) a b2 c2 2(ab bc ca) e) a b c2 2a(ab2 a c 1) f) g) a 2(1 b2 ) b2(1 c2 ) c2(1 a ) 6abc h) a b2 c2 d e2 a(b c d e) a2 b c ab ac 2bc Gi i a) a b c ab bc ca 2a 2b 2c 2ab 2bc 2ca a 2ab b b 2bc c a 2ca c 2 a b b c a c Luụn ỳng v i m i a, b, c R => PCM b) a b ab a b a b ab a b a b ab a b a b a b a b a 1 b a 12 a 1 b b Luụn ỳng v i m i a, b R => PCM c) a b c 2(a b c) a 2a b 2b c 2c Hocmai.vn Ngụi tr ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 1- Khoỏ h c Toỏn 10 - Th y L u Huy Th ng Chuyờn 04 B t ng th c v b t ph ng trỡnh a 12 b c 12 Luụn ỳng v i m i a, b,c R => PCM d) a b c 2(ab bc ca ) a b c 2ab 2bc 2ca a b c Luụn ỳng v i m i a, b,c R => PCM e) a b c 2a(ab a c 1) a 2a 2b b c 2ac 2a 2a a b c a a 12 Luụn ỳng v i m i a, b,c R => PCM f) a2 b c ab ac 2bc a2 a b c b 2bc c a2 a b c b c a b c Luụn ỳng v i m i a, b,c R => PCM g) a (1 b ) b (1 c ) c (1 a ) 6abc a a 2b b b 2c c c 2a 6abc a 2abc b 2c b 2bac a 2c c 2cba b 2a 2 a bc b ac c ab Luụn ỳng v i m i a, b,c R => PCM h) a b c d e a(b c d e) a2 a2 a2 a2 2 ab b ac c ad d ae e 4 4 2 2 a a a a b c d e ` Luụn ỳng v i m i a, b,c,d,e R => PCM Hocmai.vn Ngụi tr ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 2- Khoỏ h c Toỏn 10 - Th y L u Huy Th ng Chuyờn 04 B t ng th c v b t ph Cho a, b, c R Ch ng minh cỏc b t ng th c sau: Baứi a b a b ; v i a, b a) b) a b a 3b ab c) a 4a d) a b c 3abc , v i a, b, c > e) a b g) ng trỡnh a2 a 2 a6 b b6 a ; v i a, b f) 1a 1 b ; v i ab 1 ab h) (a b5 )(a b) (a b )(a b2 ) ; v i ab > Gi i a) a b3 a b a b a 3a b 3ab b 2 3a 3b3 3a b 3ab 3 a b a b ab 8 a b a b a, b a b a b a, b b) a b a 3b ab a a b b a b a b a b a b a ab b Luụn ỳng v i m i a,b R => PCM c) a 4a a 4a a 1a a a a 12 a 2a Luụn ỳng v i m i a R => PCM d) a b c 3 a3b 3c 3abc a,b,c e) Hocmai.vn Ngụi tr ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 3- Khoỏ h c Toỏn 10 - Th y L u Huy Th ng Chuyờn 04 B t ng th c v b t ph ng trỡnh a b6 a b8 b2 a a 2b a 6b a 2b a b a b4 a a b b a b ; v i a, b a b a b a b a a 2b b Luụn ỳng v i m i a, b,c R => PCM f) 1 2 1a b ab 1 1 1a ab b ab a b a b(a b) a ab b ab a b b a ab b a a b b ba a ab ab b a a b ab 1a b ab b a a b ab ab b a Luụn ỳng v i m i v i ab 1=> PCM g) a2 a2 2 a2 a2 a2 luụn ỳng v i m i a=> pcm h) (a b )(a b) (a b )(a b ) a b a 5b ab a b a 4b a 2b a 5b ab a 4b a 2b a 4b a b ab a b a 4b ab a b Hocmai.vn Ngụi tr ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 4- Khoỏ h c Toỏn 10 - Th y L u Huy Th ng Chuyờn 04 B t ng th c v b t ph ng trỡnh ab a b a b ab a ab b a b Luụn ỳng v i m i a.b =>pcm Cho a, b, c, d R Ch ng minh r ng a b 2ab (1) p d ng ch ng minh cỏc b t ng th c Baứi sau: a) a b c d 4abcd b) (a2 1)(b2 1)(c2 1) 8abc c) (a 4)(b2 4)(c2 4)(d 4) 256abcd Gi i Cú a b2 a b2 2ab a b2 2ab a) a b 2a 2b c d 2c 2d ] a b c d 2a 2b 2c 2d 2a 2b 2c 2d 4abcd b) a 2a 2 2 b 2b (a 1)(b 1)(c 1) 8abc c 2c c) a a 4a b b 4b c c 4c 2 (a 4)(b 4)(c 4)(d 4) 256abcd d d 4d Baứi Cho a, b, c Ch ng minh cỏc b t ng th c sau: a) (a b)(b c)(c a ) 8abc b) (a b c)(a b2 c2 ) 9abc c) (1 a )(1 b)(1 c) abc d) bc ca ab a b c ; v i a, b, c > a b c e) a 2(1 b2 ) b2(1 c2 ) c2(1 a ) 6abc f) ab bc ca a b c ; v i a, b, c > a b b c c a Hocmai.vn Ngụi tr ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 5- Khoỏ h c Toỏn 10 - Th y L u Huy Th g) ng Chuyờn 04 B t ng th c v b t ph ng trỡnh a b c ; v i a, b, c > b c c a a b Gi i a) (a b)(b c)(c a) ab bc ca a 2b2c2 8abc b) (a b c)(a b2 c2 ) 3 abc 3 a 2b2c2 9abc (1 a )(1 b)(1 c) a b c ab bc ac abc c) 13 abc 3 abc abc abc 3 d) bc ca ab a b c a b c bc ca c 2c a b ; v i a, b, c > bc ca ab ca ab a 2a 2a 2b ac a b c a b c b c bc ab b 2a a c e) a2 (1 b2 ) b2 (1 c ) c (1 a2 ) abc a2 a2 b2 b2 b2 c c c a2 6abc a2 a2 b2 b2 b2 c c c a2 Đ pcm a b6 c 6abc f) v i a, b, c > tacó ab ab tương tự ta có ab ; bc c b ac ac ; cb ca ab bc ca ab cb ac abc a b bc ca 4 2 a+b 4ab g) a b c ; v i a, b, c > b c c a a b a b c bc ca a b a b c 1 bc ca ab 1 a b c b c c a a b 2 Hocmai.vn Ngụi tr ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 6- Khoỏ h c Toỏn 10 - Th y L u Huy Th ng Chuyờn 04 B t ng th c v b t ph ng trỡnh có b c) (c a) (a b b c c a a b b c c a a b 1 b c c a a b Đ PCM Baứi Cho a, b, c > Ch ng minh cỏc b t ng th c sau: 1 a) (a b c ) (a b c)2 a b c b) 3(a b3 c3 ) (a b c)(a b2 c2 ) c) 9(a b c ) (a b c)3 Gi i a) 1 (a b3 c3 ) (a b c)2 a b c a a b b c3 c3 a b c a b c2 2ab 2bc 2ac b c a c a b 3 3 3 a a b b c c 2ab 2bc 2ac b c a c a b Thật ta có 2 a3 b3 a b3 a c3 a c3 c3 b c3 b 2ab ; 2ac ; 2bc b a b a c a c a b c b c a a b b c3 c3 2ab 2bc 2ac b c a c a b b) 3(a b c3 ) (a b c)(a b c2 ) 3a 3b 3c3 a b3 c3 ab ac2 ba bc2 ca cb 2a 2b 2c3 ab ac2 ba bc2 ca cb ta có a b a ba ab b a b ab a b ab a c3 a ca ac c2 a c ac a c ac2 b c3 b cb bc c2 b c bc b c bc2 2a 2b 2c3 ab ac2 ba bc2 ca cb c) Hocmai.vn Ngụi tr ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 7- Khoỏ h c Toỏn 10 - Th y L u Huy Th ng Chuyờn 04 B t ng th c v b t ph ng trỡnh 9(a b3 c3 ) (a b c)3 a b3 c3 a b c3 3a b 3a c 3b a 3b c 3c2a 3c2 b 6abc a b c3 3a b 3a c 3b a 3b c 3c2a 3c b 6abc tacó 2a 2b 2c3 ab ac2 ba bc2 ca cb theo CM(b) a b3 c3 3a b 3a c 3b a 3b c 3c2 a 3c2 b c ó a b3 c3 2.3 a b c3 6abc 8a b c3 3a b 3a c 3b a 3b c 3c2a 3c2 b 6abc Đ pcm Baứi Cho a, b > Ch ng minh 1 (1) p d ng ch ng minh cỏc B T sau: a b a b a) 1 1 1 ; v i a, b, c > a b b c c a a b c b) 1 1 1 ; v i a, b, c > 2a b c a 2b c a b 2c a b b c c a c) Cho a, b, c > 0tho d) 1 1 1 Ch ng minh: 2a b c a 2b c a b 2c a b c ab bc ca a b c ; v i a, b, c > a b b c c a e) Cho x, y, z > 0tho x 2y 4z 12 Ch ng minh: 2xy 8yz 4xz x 2y 2y 4z 4z x f) Cho a, b, c l di ba c nh c a m t tam giỏc, p l n a chu vi Ch ng minh r ng: 1 1 1 a b c p a p b p c Gi i a b ab a b 4ab a b 1 ab a b a b a b a v i a, b, c > 1 a b a b 1 b c b c 1 a c a c Hocmai.vn Ngụi tr 1 1 1 4 a b c b a c a b c b a c ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 8- Khoỏ h c Toỏn 10 - Th y L u Huy Th ng Chuyờn 04 B t ng th c v b t ph ng trỡnh 1 1 1 a b b c c a a b c b v i a, b, c > 1 a b b c a 2b c 1 a b a c 2a b c 1 b c a c a b 2c 1 1 1 4 a b b c a b b c a b b c a 2b c 2a b c a b 2c 1 1 1 2a b c a 2b c a b 2c a b b c c a c v i a, b, c > v theo cõu b, a ta cú 1 1 1 2a b c a 2b c a b 2c a b b c a c 1 a b c d v i a, b, c > ta cú ab a b a b bc c b ab bc ca a b c b a c c b a b b c c a 4 4 ac a c a c ab bc ca a b c a b b c c a e) v i x, y, z > 2xy x 2y x 2y 4.x.2y x 2y 8xy 2x 4y 4z 2y 4.2y.4z 4.8yz 2y 4z 4xz x 4z 4z x 4.4z.x 4.4xz x 4z 2xy 8yz 4xz x 2y 2y 4z x 4z x 2y 4z x 2y 2y 4z 4z x 4 có x 2y 4z 12 2xy 8yz 4xz x 2y 2y 4z 4z x Hocmai.vn Ngụi tr ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 9- Khoỏ h c Toỏn 10 - Th y L u Huy Th ng Chuyờn 04 B t ng th c v b t ph ng trỡnh f) cú a, b,c >0 v y 1 4 p a p b 2p a b c 1 4 p b p c 2p b c a 1 4 p a p c 2p a c b 1 1 1 2 a b c p a p b p c 1 1 1 a b c p a p b p c Baứi Cho a, b, c > Ch ng minh 1 (1) p d ng ch ng minh cỏc B T sau: a b c a b c 1 a) (a b2 c2 ) a b b c c a (a b c) b) Cho x, y, z > 0tho x y z Tỡm GTLN c a bi u th c: P = x y z x y z c) Cho a, b, c > 0tho a b c Tỡm GTNN c a bi u th c: P= a 2bc b 2ac c 2ab d) Cho a, b, c > 0tho a b c Ch ng minh: a b2 c2 1 30 ab bc ca Gi i với a,b,c>0 ta có 1 1 3 ;a b c 3 abc a b c abc 1 a b b a b c 1 a b c a bc a) V i a,b,c>0 ta cú 1 a b b c c a co ' a b b c c a 1 a b b c c a a b c 2 1 a b c (a b c ) a b b c c a a b c 2 Hocmai.vn Ngụi tr ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 10- Khoỏ h c Toỏn 10 - Th y L u Huy Th ng Chuyờn 04 B t ng th c v b t ph ng trỡnh V y ta c n CM a b2 c2 a b c (a b c) Th t v y a b2 c2 a b c (a b c) a b c a b c 2a 2b 2c 2ab 2bc 2ca a b b c c a 2 Luụn ỳng v i m i a,b,c>0 => pcm b) x y z 1 1 1 x y z x y z x y z 1 x y z voi ' : x y z co ' x y z 1 1 x y z x y z 1 => x y z x y z 4 D u = x y v ch x y z V y GTLN=3/4 t i x=y=z=1/3 c) 1 a 2bc b 2ac c 2ab a 2bc b 2ac c 2ab 1 a b c a 2bc b 2ac c 2ab Vỡ a b c nờn ta cú a 2bc b 2ac c 2ab a b c V y GTNN=9 t i a=b=c=1/3 d) Hocmai.vn Ngụi tr ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 11- Khoỏ h c Toỏn 10 - Th y L u Huy Th ng Chuyờn 04 B t ng th c v b t ph ng trỡnh 1 1 1 1 2 2 2 ab bc ca a b c 3ab 3bc 3ca 3ab 3bc 3ca a b c 1 1 42 16 * 2 2 2 3ab 3bc 3ca a b c 3ab 3bc 3ca a b c ab bc ca a b c a b c ab bc ca a b2 c2 ab bc ca a b2 c2 ab bc ca có a b b c c a với a,b,c>0 2 =>ab bc ca Từ * 1 1 16 16 12 * * 2 3ab 3bc 3ca a b c ab bc ca a b c Mặtkháctacó 2 2 1 3.3 2 3ab 3bc 3ca ab bc ca a bc 1 có a+b+c 3 abc abc a b2 c2 2 18* * * 3ab 3bc 3ca Từ * * * * * ta 1 1 30 2 ab bc ca a b c Baứi p d ng B T Cụsi tỡm GTNN c a cỏc bi u th c sau: a) y x 18 ; x x b) y x ; x x c) y 3x ; x x d) y x ;x 2x x3 x ; x e) y 1x x f) y x 4x ;x0 g) y x h) y x x2 ;x x3 ;x0 Gi i Hocmai.vn Ngụi tr ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 12- Khoỏ h c Toỏn 10 - Th y L u Huy Th ng Chuyờn 04 B t ng th c v b t ph ng trỡnh a) với x ta có x 18 x 18 x x Min y x=6 b) với x y x x x x x x x 2 x 2 x dấu "=" xảy x=3 x Vậy Min y= x=3 c)x y y 3x 3x 3x 1 3x 1 3 ; x x x 2 x 2 dấu "=" xảy Vậy Min y= 6d) x y 3x x= x 3 x= x 2x 5 2x 2x 6 dấu "=" xảy Vậy Min y=2 2x 30 2x-1= 30 x 2x 30 x= 6 e) x 51 x x x 5 x x x x x 5(1 x) dấu "=" xảy x =51 x x x x y Vậy Min y=2 x= 5 f)x x3 1 x x 1 x 33 2 x x x x dấu "=" xảy x3 =2 x 2 x y Vậy Min y=33 Hocmai.vn Ngụi tr x=3 ng chung c a h c trũ Vi t T ng i t v n: 1900 58-58-12 - Trang | 13- Khoỏ h c Toỏn 10 - Th y L u Huy Th ng Chuyờn 04 B t ng th c v b t ph ng trỡnh g)x x 4x 4 x4 48 y x x x dấu "=" xảy x =2 x 2 x Vậy Min y=33 x=3 h)x x2 x2 x2 1 yx 5 3 x 27 x x x x =3 x dấu "=" xảy x Vậy Min y=5 x= 27 Baứi p d ng B T Cụsi tỡm GTLN c a cỏc bi u th c sau: a) y (x 3)(5 x ); x c) y (x 3)(5 2x ); x e) y (6x 3)(5 2x ); b) y x(6 x ); x 5 x 2 d) y (2x 5)(5 x ); f) y x x2 ;x x g) y x2 x Gi i x,y x+y 2xy => x y 4xy x y xy a)y (x 3)(5 x) Với x x+3 0;5 x x x y 16 dấu "=" xảy x+3=5-x x Vậy Max y=16 x=1 b)Với 0