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X2 t08 04 inequality techniques (2012)

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X2 t08 04 inequality techniques (2012)

  1. 1. Types of Proof
  2. 2. Types of Proof1. Deductive Proof Start with known facts and deduce what you are trying to prove.
  3. 3. Types of Proof1. Deductive Proof Start with known facts and deduce what you are trying to prove.2. Inductive Proof Assume what you are trying to prove and induce a solution.
  4. 4. Types of Proof1. Deductive Proof Start with known facts and deduce what you are trying to prove.2. Inductive Proof Assume what you are trying to prove and induce a solution.3. Proof by Contradiction Assume the opposite of what you are trying to prove and create a contradiction. Contradiction means the original assumption is incorrect, therefore the opposite must be true.
  5. 5. Inequality Techniques
  6. 6. Inequality Techniques To prove x  y, it can be easier to prove x  y  0
  7. 7. Inequality Techniques To prove x  y, it can be easier to prove x  y  0 p2  q2e.g. i 1995 Prove pq  2
  8. 8. Inequality Techniques To prove x  y, it can be easier to prove x  y  0 p2  q2e.g. i 1995 Prove pq  2 p2  q2  pq 2
  9. 9. Inequality Techniques To prove x  y, it can be easier to prove x  y  0 p2  q2e.g. i 1995 Prove pq  2 p2  q2 p 2  2 pq  q 2  pq  2 2
  10. 10. Inequality Techniques To prove x  y, it can be easier to prove x  y  0 p2  q2e.g. i 1995 Prove pq  2 p2  q2 p 2  2 pq  q 2  pq  2 2  p  q 2  2 0
  11. 11. Inequality Techniques To prove x  y, it can be easier to prove x  y  0 p2  q2e.g. i 1995 Prove pq  2 p2  q2 p 2  2 pq  q 2  pq  2 2  p  q2  2 0 p2  q2   pq 2
  12. 12. Inequality Techniques To prove x  y, it can be easier to prove x  y  0 p2  q2e.g. i 1995 Prove pq  2 p q 2 2 p  2 pq  q 2 2 p2  q2  pq  OR Assume pq  2 2 2  p  q2  2 0 p2  q2   pq 2
  13. 13. Inequality Techniques To prove x  y, it can be easier to prove x  y  0 p2  q2e.g. i 1995 Prove pq  2 p q 2 2 p  2 pq  q 2 2 p2  q2  pq  OR Assume pq  2 2 2  p  q2 2 pq  p 2  q 2  2 0  p 2  2 pq  q 2 0 0  ( p  q) 2 p2  q2   pq 2
  14. 14. Inequality Techniques To prove x  y, it can be easier to prove x  y  0 p2  q2e.g. i 1995 Prove pq  2 p q 2 2 p  2 pq  q 2 2 p2  q2  pq  OR Assume pq  2 2 2  p  q2 2 pq  p 2  q 2  2 0  p 2  2 pq  q 2 0 0  ( p  q) 2 p2  q2   pq 2 But ( p  q) 2  0
  15. 15. Inequality Techniques To prove x  y, it can be easier to prove x  y  0 p2  q2e.g. i 1995 Prove pq  2 p q 2 2 p  2 pq  q 2 2 p2  q2  pq  OR Assume pq  2 2 2  p  q2 2 pq  p 2  q 2  2 0  p 2  2 pq  q 2 0 0  ( p  q) 2 p2  q2   pq 2 But ( p  q) 2  0 p2  q2  pq  2
  16. 16. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac
  17. 17. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac a  b   0 2
  18. 18. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac a  b   0 2 a 2  2ab  b 2  0
  19. 19. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac a  b   0 2 a 2  2ab  b 2  0  a 2  b 2  2ab
  20. 20. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac a  b   0 2 a 2  2ab  b 2  0  a 2  b 2  2ab a 2  c 2  2ac b 2  c 2  2bc
  21. 21. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac a  b   0 2 a 2  2ab  b 2  0  a 2  b 2  2ab a 2  c 2  2ac b 2  c 2  2bc 2a 2  2b 2  2c 2  2ab  2ac  2bc
  22. 22. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac a  b   0 2 a 2  2ab  b 2  0  a 2  b 2  2ab a 2  c 2  2ac b 2  c 2  2bc 2a 2  2b 2  2c 2  2ab  2ac  2bc a 2  b 2  c 2  ab  ac  bc
  23. 23. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac a  b   0 2 a 2  2ab  b 2  0  a 2  b 2  2ab a 2  c 2  2ac b 2  c 2  2bc 2a 2  2b 2  2c 2  2ab  2ac  2bc a 2  b 2  c 2  ab  ac  bc 1b) If a  b  c  1, prove ab  ac  bc  3
  24. 24. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac a  b   0 2 a 2  2ab  b 2  0  a 2  b 2  2ab a 2  c 2  2ac b 2  c 2  2bc 2a 2  2b 2  2c 2  2ab  2ac  2bc a 2  b 2  c 2  ab  ac  bc 1b) If a  b  c  1, prove ab  ac  bc  3 a  b  c  a  b  c   2ab  ac  bc  2 2 2 2
  25. 25. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac a  b   0 2 a 2  2ab  b 2  0  a 2  b 2  2ab a 2  c 2  2ac b 2  c 2  2bc 2a 2  2b 2  2c 2  2ab  2ac  2bc a 2  b 2  c 2  ab  ac  bc 1b) If a  b  c  1, prove ab  ac  bc  3 a  b  c  a  b  c   2ab  ac  bc  2 2 2 2  a  b  c   2ab  ac  bc   ab  ac  bc 2
  26. 26. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac a  b   0 2 a 2  2ab  b 2  0  a 2  b 2  2ab a 2  c 2  2ac b 2  c 2  2bc 2a 2  2b 2  2c 2  2ab  2ac  2bc a 2  b 2  c 2  ab  ac  bc 1b) If a  b  c  1, prove ab  ac  bc  3 a  b  c  a  b  c   2ab  ac  bc  2 2 2 2  a  b  c   2ab  ac  bc   ab  ac  bc 2 3ab  ac  bc   a  b  c  2
  27. 27. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac a  b   0 2 a 2  2ab  b 2  0  a 2  b 2  2ab a 2  c 2  2ac b 2  c 2  2bc 2a 2  2b 2  2c 2  2ab  2ac  2bc a 2  b 2  c 2  ab  ac  bc 1b) If a  b  c  1, prove ab  ac  bc  3 a  b  c  a  b  c   2ab  ac  bc  2 2 2 2  a  b  c   2ab  ac  bc   ab  ac  bc 2 3ab  ac  bc   a  b  c  2 3ab  ac  bc   1 1 ab  ac  bc  3
  28. 28. 1c) Prove a  b  c   3 abc 3
  29. 29. 1c) Prove a  b  c   3 abc 3 a 2  b 2  c 2  ab  ac  bc
  30. 30. 1c) Prove a  b  c   3 abc 3 a 2  b 2  c 2  ab  ac  bc a 2  b 2  c 2  ab  ac  bc  0
  31. 31. 1c) Prove a  b  c   3 abc 3 a 2  b 2  c 2  ab  ac  bc a 2  b 2  c 2  ab  ac  bc  0 a  b  c a 2  b 2  c 2  ab  ac  bc   0
  32. 32. 1c) Prove a  b  c   3 abc 3 a 2  b 2  c 2  ab  ac  bc a 2  b 2  c 2  ab  ac  bc  0 a  b  c a 2  b 2  c 2  ab  ac  bc   0 a 3  ab 2  ac 2  a 2b  a 2 c  abc  a 2b  b3  bc 2  ab 2  abc  b 2 c  a 2 c  b 2 c  c 3  abc  ac 2  bc 2  0
  33. 33. 1c) Prove a  b  c   3 abc 3 a 2  b 2  c 2  ab  ac  bc a 2  b 2  c 2  ab  ac  bc  0 a  b  c a 2  b 2  c 2  ab  ac  bc   0 a 3  ab 2  ac 2  a 2b  a 2 c  abc  a 2b  b3  bc 2  ab 2  abc  b 2 c  a 2 c  b 2 c  c 3  abc  ac 2  bc 2  0 a 3  b3  c3  3abc  0
  34. 34. 1c) Prove a  b  c   3 abc 3 a 2  b 2  c 2  ab  ac  bc a 2  b 2  c 2  ab  ac  bc  0 a  b  c a 2  b 2  c 2  ab  ac  bc   0 a 3  ab 2  ac 2  a 2b  a 2 c  abc  a 2b  b3  bc 2  ab 2  abc  b 2 c  a 2 c  b 2 c  c 3  abc  ac 2  bc 2  0 a 3  b3  c3  3abc  0 a  b  c   abc 1 3 3 3 3
  35. 35. 1c) Prove a  b  c   3 abc 3 a 2  b 2  c 2  ab  ac  bc a 2  b 2  c 2  ab  ac  bc  0 a  b  c a 2  b 2  c 2  ab  ac  bc   0 a 3  ab 2  ac 2  a 2b  a 2 c  abc  a 2b  b3  bc 2  ab 2  abc  b 2 c  a 2 c  b 2 c  c 3  abc  ac 2  bc 2  0 a 3  b3  c3  3abc  0 a  b  c   abc 1 3 3 3 3 1 1 1let a  a , b  b , c  c 3 3 3
  36. 36. 1c) Prove a  b  c   3 abc 3 a 2  b 2  c 2  ab  ac  bc a 2  b 2  c 2  ab  ac  bc  0 a  b  c a 2  b 2  c 2  ab  ac  bc   0 a 3  ab 2  ac 2  a 2b  a 2 c  abc  a 2b  b3  bc 2  ab 2  abc  b 2 c  a 2 c  b 2 c  c 3  abc  ac 2  bc 2  0 a 3  b3  c3  3abc  0 a  b  c   abc 1 3 3 3 3 1 1 1let a  a , b  b , c  c 3 3 3 1 1 1 1 a  b  c   a 3b 3 c 3 3
  37. 37. 1c) Prove a  b  c   3 abc 3 a 2  b 2  c 2  ab  ac  bc a 2  b 2  c 2  ab  ac  bc  0 a  b  c a 2  b 2  c 2  ab  ac  bc   0 a 3  ab 2  ac 2  a 2b  a 2 c  abc  a 2b  b3  bc 2  ab 2  abc  b 2 c  a 2 c  b 2 c  c 3  abc  ac 2  bc 2  0 a 3  b3  c3  3abc  0 a  b  c   abc 1 3 3 3 3 1 1 1let a  a , b  b , c  c 3 3 3 1 1 1 1 a  b  c   a 3b 3 c 3 3 1 a  b  c   3 abc 3
  38. 38. Arithmetic Mean  Geometric Mean a1  a2    an n  a1a2  an n
  39. 39. Arithmetic Mean  Geometric Mean a1  a2    an n  a1a2  an nd) Suppose 1  x 1  y 1  z   8, prove xyz  1
  40. 40. Arithmetic Mean  Geometric Mean a1  a2    an n  a1a2  an nd) Suppose 1  x 1  y 1  z   8, prove xyz  1 1  x 1  y 1  z   8 1  x  y  xy  z  xz  yz  xyz  8
  41. 41. Arithmetic Mean  Geometric Mean a1  a2    an n  a1a2  an nd) Suppose 1  x 1  y 1  z   8, prove xyz  1 1  x 1  y 1  z   8 1  x  y  xy  z  xz  yz  xyz  8 1  x  y  z   3 xyz AM  GM 3
  42. 42. Arithmetic Mean  Geometric Mean a1  a2    an n  a1a2  an nd) Suppose 1  x 1  y 1  z   8, prove xyz  1 1  x 1  y 1  z   8 1  x  y  xy  z  xz  yz  xyz  8 1  x  y  z   3 xyz AM  GM 3 x  y  z  33 xyz
  43. 43. Arithmetic Mean  Geometric Mean a1  a2    an n  a1a2  an nd) Suppose 1  x 1  y 1  z   8, prove xyz  1 1  x 1  y 1  z   8 1  x  y  xy  z  xz  yz  xyz  8 1  x  y  z   3 xyz AM  GM 3 x  y  z  33 xyz xy  yz  xz  33  xy  yz  xz 
  44. 44. Arithmetic Mean  Geometric Mean a1  a2    an n  a1a2  an nd) Suppose 1  x 1  y 1  z   8, prove xyz  1 1  x 1  y 1  z   8 1  x  y  xy  z  xz  yz  xyz  8 1  x  y  z   3 xyz AM  GM 3 x  y  z  33 xyz xy  yz  xz  33  xy  yz  xz  xy  yz  xz  33 x 2 y 2 z 2
  45. 45. Arithmetic Mean  Geometric Mean a1  a2    an n  a1a2  an nd) Suppose 1  x 1  y 1  z   8, prove xyz  1 1  x 1  y 1  z   8 1  x  y  xy  z  xz  yz  xyz  8 1  x  y  z   3 xyz AM  GM 3 x  y  z  33 xyz xy  yz  xz  33  xy  yz  xz  xy  yz  xz  33 x 2 y 2 z 2 xy  yz  xz  3 xyz  2 3
  46. 46. 1  x  y  z  xy  xz  yz  xyz  8
  47. 47. 1  x  y  z  xy  xz  yz  xyz  8 1  3 xyz  3 xyz   xyz  8 2 3 3
  48. 48. 1  x  y  z  xy  xz  yz  xyz  8 1  3 xyz  3 xyz   xyz  8 2 3 31  3 xyz  3 xyz    xyz   8 2 3 3 3 3
  49. 49. 1  x  y  z  xy  xz  yz  xyz  8 1  3 xyz  3 xyz   xyz  8 2 3 31  3 xyz  3 xyz    xyz   8 2 3 3 3 3 1  3 xyz   8 3
  50. 50. 1  x  y  z  xy  xz  yz  xyz  8 1  3 xyz  3 xyz   xyz  8 2 3 31  3 xyz  3 xyz    xyz   8 2 3 3 3 3 1  3 xyz   8 3 1  3 xyz  2
  51. 51. 1  x  y  z  xy  xz  yz  xyz  8 1  3 xyz  3 xyz   xyz  8 2 3 31  3 xyz  3 xyz    xyz   8 2 3 3 3 3 1  3 xyz   8 3 1  3 xyz  2 3 xyz  1
  52. 52. 1  x  y  z  xy  xz  yz  xyz  8 1  3 xyz  3 xyz   xyz  8 2 3 31  3 xyz  3 xyz    xyz   8 2 3 3 3 3 1  3 xyz   8 3 1  3 xyz  2 3 xyz  1 xyz  1
  53. 53. OR 1  x   2 x AM  GM
  54. 54. OR 1  x   2 x AM  GM 1  y   2 y 1  z   2 z
  55. 55. OR 1  x   2 x AM  GM 1  y   2 y 1  z   2 z (1  x)(1  y ) 1  z   2 x 2 y 2 z  8 xyz
  56. 56. OR 1  x   2 x AM  GM 1  y   2 y 1  z   2 z (1  x)(1  y ) 1  z   2 x 2 y 2 z  8 xyz 8  8 xyz
  57. 57. OR 1  x   2 x AM  GM 1  y   2 y 1  z   2 z (1  x)(1  y ) 1  z   2 x 2 y 2 z  8 xyz 8  8 xyz 1  1 xyz xyz  1 xyz  1
  58. 58. 9 2 2 2 1 1 1 iii  Prove       abc ab bc a c a b c
  59. 59. 9 2 2 2 1 1 1 iii  Prove       abc ab bc a c a b c 1 1 1 a  b      2 ab  2 a b ab 4
  60. 60. 9 2 2 2 1 1 1 iii  Prove       abc ab bc a c a b c 1 1 1 a  b      2 ab  2 a b ab 4 1 1 4   a b ab
  61. 61. 9 2 2 2 1 1 1 iii  Prove       abc ab bc a c a b c 1 1 1 a  b      2 ab  2 a b ab 4 1 1 4   a b ab 1 1 4   b c bc 1 1 4   a c ac
  62. 62. 9 2 2 2 1 1 1 iii  Prove       abc ab bc a c a b c 1 1 1 a  b      2 ab  2 a b ab 4 1 1 4   a b ab 1 1 4   b c bc 1 1 4   a c ac2 2 2 4 4 4     a b c ab bc ac
  63. 63. 9 2 2 2 1 1 1 iii  Prove       abc ab bc a c a b c 1 1 1 a  b      2 ab  2 a b ab 4 1 1 4   a b ab 1 1 4   b c bc 1 1 4   a c ac2 2 2 4 4 4     a b c ab bc ac1 1 1 2 2 2     a b c ab bc ac
  64. 64. 9 2 2 2 1 1 1 iii  Prove       abc ab bc a c a b c 1 1 1 1 1 1 1 a  b      2 ab  2  a  b  c       3 3 abc  3 3 a b ab a b c abc 4 9 1 1 4   a b ab 1 1 4   b c bc 1 1 4   a c ac2 2 2 4 4 4     a b c ab bc ac1 1 1 2 2 2     a b c ab bc ac
  65. 65. 9 2 2 2 1 1 1 iii  Prove       abc ab bc a c a b c 1 1 1 1 1 1 1 a  b      2 ab  2  a  b  c       3 3 abc  3 3 a b ab a b c abc 4 9 1 1 4 1 1 1 9      a b ab a b c abc 1 1 4   b c bc 1 1 4   a c ac2 2 2 4 4 4     a b c ab bc ac1 1 1 2 2 2     a b c ab bc ac
  66. 66. 9 2 2 2 1 1 1 iii  Prove       abc ab bc a c a b c 1 1 1 1 1 1 1 a  b      2 ab  2  a  b  c       3 3 abc  3 3 a b ab a b c abc 4 9 1 1 4 1 1 1 9      a b ab 1 1a b 1 c a  b  c 9 1 1   4    b c bc a  b b  c a  c 2a  b  c 1 1 4 2 2 2 9      a c ac ab bc ac abc2 2 2 4 4 4     a b c ab bc ac1 1 1 2 2 2     a b c ab bc ac
  67. 67. 9 2 2 2 1 1 1 iii  Prove       abc ab bc a c a b c 1 1 1 1 1 1 1 a  b      2 ab  2  a  b  c       3 3 abc  3 3 a b ab a b c abc 4 9 1 1 4 1 1 1 9      a b ab 1 1a b 1 c a  b  c 9 1 1   4    b c bc a  b b  c a  c 2a  b  c 1 1 4 2 2 2 9      a c ac ab bc ac abc2 2 2 4 4 4     a b c ab bc ac1 1 1 2 2 2     a b c ab bc ac 9 2 2 2 1 1 1       abc ab bc ac a b c
  68. 68. 9 2 2 2 1 1 1 iii  Prove       abc ab bc a c a b c 1 1 1 1 1 1 1 a  b      2 ab  2  a  b  c       3 3 abc  3 3 a b ab a b c abc 4 9 1 1 4 1 1 1 9      a b ab 1 1a b 1 c a  b  c 9 1 1   4    b c bc a  b b  c a  c 2a  b  c 1 1 4 2 2 2 9      a c ac ab bc ac abc2 2 2 4 4 4 Inequalities Sheet     a b c ab bc ac1 1 1 2 2 2 Exercise 10D     a b c ab bc ac Note: Cambridge 8H (Book 1); 28 9 2 2 2 1 1 1       abc ab bc ac a b c

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