www.facebook.com/groups/TaiLieuOnThiDaiHoc01 24H H C TOÁN - CHI N TH NG CÂU PHÂN LO I Giáo viên Đoàn Trí Dũng Hà H u H i BÀI 9: K THU T D N V M T BI N B NG CAUCHY Bài 1: Cho s th c d ng a, b, c Tìm giá tr l n nh t c a bi u th c: 1 P    a  b  c b  c  a c  a  b  a  b  c 2 Áp d ng b t đ ng th c Cauchy    SCHWARZ Schwarz cho s th c d ng a, b, c :     a2  b  c  b  c  a  b  c      1 b  c 1 c  a 1 a  b   ,  ,   b  c  a 1  c  a    a  b  c   2 a  b  c  a  b  c  b  c  a  a  b  c  c  a  b  a  b  c 2   c  a  b 1  a  b    a  b  c   Do a bc V y P  b ca  2a  b  c a  b  c   c ab a  b  c   2a  b  c a  b  c  a  b  c  a  b  c 2 P Đ t t  a  b  c , t   0;   xét hàm s f t    Ta có P  f  a  b  c  t t2 L p b ng bi n thiên ta có P  f  a  b  c   f    Bài 2: Cho s th c d Đ ng th c x y a  b  c  ng a, b, c th a mãn a2  b2  c  a  b  c Tìm giá tr l n nh t c a bi u th c: 1 15  2   P 2 a  b  b  c  c  a   a  b  c 2 Áp d ng b t đ ng th c Cauchy Schwarz cho s th c d       ng a, b, c :     a2  b2  1   c  a  b  c     2  b  c  1   a  a  b  c   c  a2  1   b2   a  b  c    a  b2    b2  c  1 a  b2    c  a2  1 b2  c    c2  a  b  c c  a2    a2  a  b  c  b2  a  b  c a  b2  c  a  b  c Chú ý r ng ta có u ki n: a2  b2  c  a  b  c Do www.facebook.com/groups/TaiLieuOnThiDaiHoc01 www.facebook.com/groups/TaiLieuOnThiDaiHoc01 P abc6 a  b  c  Đ t t  a  b  c , t   0;   xét hàm s 15 2a  b  c f t    a  b  c  a  b  c 2  t 2t L p b ng bi n thiên ta có P  f  a  b  c   f    Bài 3: Cho s th c d P Đ ng th c x y a  b  c  ng a, b, c th a mãn a2  b2  c  a  b  c Tìm giá tr l n nh t c a bi u th c: 1 P    a  b  b  c  c  a   a  b  c 2 Áp d ng b t đ ng th c Cauchy Schwarz cho s th c d       ng a, b, c :     a2  b  1  b  c  a  b  c     2  b  c  1  c  a  a  b  c   c  a  1  a  b2   a  b  c    a2  b    b2  c  1 a2  b    c2  a  1 b2  c   c2  b  a  b  c  c2  a    a2  c  a  b  c  b2  a  a  b  c a2  b2  c  a  b  c  a  b  c Chú ý r ng ta có u ki n: a2  b2  c  a  b  c Do P 2a  b  c  a  b  c Đ t t  a  b  c , t   0;   xét hàm s  a  b  c f t   P  a  b  c  a  b  c 2  t t2 L p b ng bi n thiên ta có P  f  a  b  c   f    Đ ng th c x y a  b  c  Bài 4: Cho s th c không âm a, b, c Tìm giá tr l n nh t c a bi u th c: P  a  2b  3c  Áp d ng b t đ ng th c Cauchy  a  2b  3c  V y P  a  2b  3c   a  b2  c 2 Schwarz cho s th c không âm a, b, c :      a2  b2  c  14 a  b2  c  a  b2  c 2     a2  b2  c 12  22  32  a  2b  3c  14 a2  b2  c    Khi P    a2  b2  c  14 a2  b2  c  14   2  www.facebook.com/groups/TaiLieuOnThiDaiHoc01  www.facebook.com/groups/TaiLieuOnThiDaiHoc01 P  a  b  c  14 2  a2  b2  c  14   a  1, b  2, c    Đ ng th c x y ch khi:  a b c    0 1 Bài 5: Cho s a, b, c  th a mãn 1    Tìm giá tr nh nh t c a bi u th c: bc 1 c  a1 ab1 P  a  b  c   ab  bc  ca  Áp d ng b t đ ng th c Cauchy    Schwarz cho s th c d ng a, b, c :      b  c a2  b  c  a  b  c     a2  b  c b2  c  a c2  a  b 1    ,  ,   c  a b  c  a  a  b  c   bc 1   2  a  b  c c  a  a  b  c a  b  a  b  c 2   a  b  c  a  b   a  b  c   a  b2  c   a  b  c  1     bc 1 c  a1 ab1 a  b  c M t khác, a2  b2  c   a  b  c  1 1   1 bc 1 c  a1 ab1 a b c      a2  b2  c   a  b  c    a  b  c   a2  b2  c   a  b  c   a2  b2  c   ab  bc  ca   a  b  c  ab  bc  ca V y P   ab  bc  ca    ab  bc  ca  P  ab  bc  ca     3 Đ ng th c x y a  b  c  Bài 6: Cho s th c x, y , z  th a mãn 1    Tìm giá tr l n nh t c a bi u th c: x y z P  x 1  y 1  z 1  Áp d ng b t đ ng th c Cauchy xyz Schwarz cho s th c x, y , z  : x 1  y 1  z 1  x  y  z x 1 y 1 z 1   x y z  1 1  x 1  y 1  z 1  x  y  z       x 1  y 1  z 1  x  y  z x y z Do P  x  y  z  xyz L p b ng bi n thiên ta đ Đ t t  x  y  z , t   3;   xét hàm s f t   t  t 9 3 c P  f a  b  c  f    Đ ng th c x y x  y  z  2 www.facebook.com/groups/TaiLieuOnThiDaiHoc01 www.facebook.com/groups/TaiLieuOnThiDaiHoc01 Bài 7: Cho s th c d ng x , y , z th a mãn x  y  z  1   Tìm giá tr l n nh t c a bi u th c: x y z y2  x2  z2   x  y  z     2 P Áp d ng b t đ ng th c Cauchy Schwarz cho s th c d ng x , y , z : y2  x2  z2  x2  y  z     xyz   2 2x 2y 2z  y2  1 1 1 x2  z2     xyz x  y  z     2 2 x y z y2  x2  z2  1    xyz x  y  z  x  y  z  2 2  Do x  y  z P xyz P P ng a, b, c Tìm giá tr nh nh t c a bi u th c: P  Áp d ng b t đ ng th c Cauchy V y  x  y  z  6x  y  z    2 3 Đ ng th c x y x  y  z  x  y  z  3    2 Bài 8: Cho s th c d V y  y2  x2  z2     xyz 2  a  b  a  b Do P    a c a  c a  bc   bc  b  c   a Schwarz cho s th c d  a  b  T   bc  b  c   a  c  a  b   a b a  c   bc  b  c    a  c  a2  bc  b  c  b   a  b  c c ng t ta có: ng a, b, c : a  b  a2  bc  bc 1 a  bc  a2  bc Áp d ng b t đ ng th c AM GM ta có: P a  bc  a2  bc  a2  bc  3 Đ ng th c x y ch a  b  c  a  bc a2  bc a  bc  P  www.facebook.com/groups/TaiLieuOnThiDaiHoc01  a2  bc