Bai 20 DABTTL PP becnuli

Khoá h c BDHSG Chuyên đ B t đ ng th c – Th y Tr n Ph BÀI 20 B T ng B t đ ng th c Bernouli NG TH C BERNOULI ÁP ÁN BÀI T P T Giáo viên: LÊ Các t p tài li u đ LUY N C VI T c biên so n kèm theo gi ng Bài 20 B t đ ng th c Bernouli thu c khóa h c B i d ng h c sinh gi i Chuyên đ B t đ ng th c – Th y Tr n Ph ng t i website Hocmai.vn c n h c tr c gi ng sau làm đ y đ t p tài li u s d ng hi u qu , b n Bài Cho x, y > 0,  ≥ Ch ng minh r ng: x   y  21  x  y   Ch ng minh: S d ng: x  y2  2xy   x  y2    x  y     x xy 2  y    2 xy Áp d ng b t đ ng th c Bernoulli (*) ta có:      2x    2x     xy 2 xy     2y        2y    2  x  y   x  y     y    x    xy xy  2  x  xy        y        2  x   xy   xy      y   2  2   xy   21  x   y  21  x  y  Bài Cho a, b > Ch ng minh r ng: a  b3  21  ab  a  b  Ch ng minh: 3 2 S d ng: a  b   a  b   a  b  ab    a  b  ab  a2  b 1 b a  b a a  b Áp d ng b t đ ng th c Bernoulli (*) ta có:  a    b  a  b       b   a  a  b    2  a3  a2   b a  b  b3  21 a   2 1  2   2  b  a  b   b   2 1  2   2    a a  b  b2  a a  b 2 ab  a  b  2   a2 b2   2 2 2       b  a  b  a  a  b    2 D u b ng x y  a = b Hocmai.vn – Ngôi tr ng chung c a h c trò Vi t T ng đài t v n: 1900 58-58-12 - Trang | - Khoá h c BDHSG Chuyên đ B t đ ng th c – Th y Tr n Ph ng B t đ ng th c Bernouli Bài Cho a, b, c, d > Tìm giá tr nh nh t c a bi u th c: S  a bcd    b cda    c da b    d a bc  Gi i B đ : a b c d    4 bcd cda da b a bc  Ch ng minh: a b c d    4 bcd cda da b a bc          a b c d  1  1  1  4 bcd cda da b a bc 1 1  a  b  c  d     16 bcd cda dab abc 1 1  P  3 a  b  c  d      16 bcd cda dab abc  1    S d ng b t đ ng th c Cơsi ta có:  a  b  c    b  c  d    c  d  a    d  a  b   4  a  b  c  b  c  d  c  d  a  d  a  b    1 1  b  c  d  c  d  a  d  a  b  a  b  c   a  b  c  b  c  d  c  d  a  d  a  b   1 1  P  3a  b  c  d     bcd cda da b a bc 1 1   b  c  d    c  d  a    d  a  b    a  b  c     bcd cda da b a bc     4  a  b  c   b  c  d  c  d  a  d  a  b      16  a  b  c  b  c  d  c  d  a  d  a  b  Áp d ng b t đ ng th c Bernoulli ta có:      b 3ac  d    c 3bd  a      c 3bd  a   d 3ca  b      d 3ca  b   a 3db  c      a 3db  c    a b c d       4     cda dab a  b  c   bcd a b c d  3    4 bcd cd a d a  b a  bc d a b c      a  b  c bcd cda d a b            3a bcd 3 3 3 3 3 3  1 3 Hocmai.vn – Ngơi tr ng chung c a h c trò Vi t T ng đài t v n: 1900 58-58-12 - Trang | - Khoá h c BDHSG Chuyên đ B t đ ng th c – Th y Tr n Ph ng B t đ ng th c Bernouli V i a = b = c = d > MinS  3 Bài Cho a, b, c đ dài c nh c a tam giác ABC Tìm giá tr nh nh t c a T  a 2b  2c  a    b 2c  2a  b    c 2a  2b  c  Gi i B đ : a b c    1 2b  2c  a 2c  2a  b 2a  2b  c  Ch ng minh: 1  3a 3b 3c   3 2b  2c  a 2c  2a  b 2a  2b  c 3a 3b 3c  1  1  1 6 2b  2c  a 2c  2a  b 2a  2b  c 1  P   2b  2c  a    2c  2a  b    2a  2b  c    2b  2c  a 2c  2a  b 2a  2b  c    2b  2c  a  2c  2a  b  2a  2b  c     Áp d ng:  2b  2c  a  2c  2a  b  2a  2b  c        s d ng b t đ ng th c Bernoulli ta có:     3a  2b  2c  a   3b  2c  2a  b   3c  2a  2b  c     2  2b 3a2c  a  3b 2 2c  2a  b  3c 2 2a  2b  c   1   1   1          a b c T   2b  2c  a  2c  2a  b  2a  2b  c  2  a b c 3    2c  2a  b 2a  2b  c  2b  2c  a 3a 3b 3c    3 2b  2c  a 2c  2a  b 2a  2b  c 2 V i a = b = c >  ∆ABC đ u MinT  31 2      1  32  31 2  x, y, z  0; x   Bài Cho 3x  4y  24 Tìm giá tr nh nh t c a S  x 3x  4y  6z  36  y z Gi i y y T gi thi t suy ra: x  1; x   2; x   z  Hocmai.vn – Ngôi tr 4 ng chung c a h c trò Vi t T ng đài t v n: 1900 58-58-12 - Trang | - Khoá h c BDHSG Chuyên đ B t đ ng th c – Th y Tr n Ph ng B t đ ng th c Bernouli S d ng b t đ ng th c Bernoulli ta có: x    y  z  x    1 2    1 2    1 2 y   2 z x   x  4 y  2.3 1.y    3  2.2 1.z   z  2    1   2    3y  2z  3 y  x 3  2  x y 2  3 z 4  2 y   2 x   z   3.4 1  2.4 4 2 z 2    2 3 2 3 4     z 2 2 2 4 3 3 2   2   2 V i x = 4, y = 3, z = MinS = 4  x  y  z   Bài Cho 3x  4y  2xy Tìm giá tr l n nh t c a S  x 2xy  3xz  4yz  3xyz  y 3 2 z Gi i T gi thi t suy  1;   2;    x x y x y z S d ng b t đ ng th c Bernoulli ta có: 4    1 x    3    1 y  2    1 z  4 3 2 3 3  3 x x  3.3.y 1    y y  3.2.z 1    z z x y 3  2 y z   3y  y  2z  z      x      y  x    z  x y z y z z    1  x   3.x 1  3.4.x 4 x 3 y z  x x 3   z y 3    x y z 3 3 y  3  V i x = 4, y = 3, z = MaxS = 3 Ngu n ng chung c a h c trò Vi t    2 Giáo viên : Lê Hocmai.vn – Ngôi tr T ng đài t v n: 1900 58-58-12 : c Vi t Hocmai.vn - Trang | -