X2 T08 04 inequality techniques

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<p> 1. Inequality Techniques 2. Inequality Techniques To prove x y, it can be easier to prove x y 0 3. Inequality TechniquesTo prove x y, it can be easier to prove x y 0p2 q2 e.g. i 1995 Prove pq 2 4. Inequality TechniquesTo prove x y, it can be easier to prove x y 0 p2 q2 e.g. i 1995 Prove pq 2 p2 q2 pq2 5. Inequality TechniquesTo prove x y, it can be easier to prove x y 0 p2 q2 e.g. i 1995 Prove pq 2 p2 q2p 2 2 pq q 2 pq 22 6. Inequality TechniquesTo prove x y, it can be easier to prove x y 0 p2 q2 e.g. i 1995 Prove pq 2 p2 q2p 2 2 pq q 2 pq 22 p q 2 2 7. Inequality TechniquesTo prove x y, it can be easier to prove x y 0 p2 q2 e.g. i 1995 Prove pq 2 p2 q2p 2 2 pq q 2 pq 22 p q 2 2 0 8. Inequality TechniquesTo prove x y, it can be easier to prove x y 0 p2 q2 e.g. i 1995 Prove pq 2 p2 q2p 2 2 pq q 2 pq 22 p q220p2 q2 pq 2 9. ii 1994 a) Prove a 2 b 2 c 2 ab bc ac 10. ii 1994 a) Prove a 2 b 2 c 2 ab bc ac a b 2 0 11. ii 1994 a) Prove a 2 b 2 c 2 ab bc ac a b 2 0 a 2 2ab b 2 0 12. ii 1994 a) Prove a 2 b 2 c 2 ab bc ac a b 2 0 a 2 2ab b 2 0 a 2 b 2 2ab 13. ii 1994 a) Prove a 2 b 2 c 2 ab bc ac a b 2 0 a 2 2ab b 2 0 a 2 b 2 2ab a 2 c 2 2ac b 2 c 2 2bc 14. ii 1994 a) Prove a 2 b 2 c 2 ab bc ac a b 2 0 a 2 2ab b 2 0 a 2 b 2 2ab a 2 c 2 2acb 2 c 2 2bc 2a 2 2b 2 2c 2 2ab 2ac 2bc 15. ii 1994 a) Prove a 2 b 2 c 2 ab bc ac a b 2 0 a 2 2ab b 2 0 a 2 b 2 2ab a 2 c 2 2acb 2 c 2 2bc 2a 2 2b 2 2c 2 2ab 2ac 2bca 2 b 2 c 2 ab ac bc 16. ii 1994 a) Prove a 2 b 2 c 2 ab bc ac a b 2 0 a 2 2ab b 2 0 a 2 b 2 2ab a 2 c 2 2acb 2 c 2 2bc 2a 2 2b 2 2c 2 2ab 2ac 2bca 2 b 2 c 2 ab ac bc1 b) If a b c 1, prove ab ac bc 3 17. ii 1994 a) Prove a 2 b 2 c 2 ab bc ac a b 2 0 a 2 2ab b 2 0 a 2 b 2 2ab a 2 c 2 2acb 2 c 2 2bc 2a 2 2b 2 2c 2 2ab 2ac 2bca 2 b 2 c 2 ab ac bc1 b) If a b c 1, prove ab ac bc 3 a b c a b c 2ab ac bc 2 2 2 2 18. ii 1994 a) Prove a 2 b 2 c 2 ab bc ac a b 2 0 a 2 2ab b 2 0 a 2 b 2 2ab a 2 c 2 2acb 2 c 2 2bc 2a 2 2b 2 2c 2 2ab 2ac 2bca 2 b 2 c 2 ab ac bc1 b) If a b c 1, prove ab ac bc 3 a b c a b c 2ab ac bc 2 2 2 2 a b c 2ab ac bc ab ac bc 2 19. ii 1994 a) Prove a 2 b 2 c 2 ab bc ac a b 2 0 a 2 2ab b 2 0 a 2 b 2 2ab a 2 c 2 2acb 2 c 2 2bc 2a 2 2b 2 2c 2 2ab 2ac 2bca 2 b 2 c 2 ab ac bc1 b) If a b c 1, prove ab ac bc 3 a b c a b c 2ab ac bc 2 2 2 2 a b c 2ab ac bc ab ac bc 23ab ac bc a b c 2 20. ii 1994 a) Prove a 2 b 2 c 2 ab bc ac a b 2 0 a 2 2ab b 2 0 a 2 b 2 2ab a 2 c 2 2acb 2 c 2 2bc 2a 2 2b 2 2c 2 2ab 2ac 2bca 2 b 2 c 2 ab ac bc1 b) If a b c 1, prove ab ac bc 3 a b c a b c 2ab ac bc 2 2 2 2 a b c 2ab ac bc ab ac bc 23ab ac bc a b c 23ab ac bc 1 1 ab ac bc 3 21. 1 c) Prove a b c 3 abc3 22. 1 c) Prove a b c 3 abc3 a 2 b 2 c 2 ab ac bc 23. 1 c) Prove a b c 3 abc3 a 2 b 2 c 2 ab ac bca 2 b 2 c 2 ab ac bc 0 24. 1 c) Prove a b c 3 abc3 a 2 b 2 c 2 ab ac bca 2 b 2 c 2 ab ac bc 0a b c a 2 b 2 c 2 ab ac bc 0 25. 1 c) Prove a b c 3 abc3 a 2 b 2 c 2 ab ac bca 2 b 2 c 2 ab ac bc 0a b c a 2 b 2 c 2 ab ac bc 0a 3 ab 2 ac 2 a 2b a 2 c abc a 2b b 3 bc 2 ab 2 abc b 2 c a 2 c b 2 c c 3 abc ac 2 bc 2 0 26. 1 c) Prove a b c 3 abc3 a 2 b 2 c 2 ab ac bca 2 b 2 c 2 ab ac bc 0a b c a 2 b 2 c 2 ab ac bc 0a 3 ab 2 ac 2 a 2b a 2 c abc a 2b b 3 bc 2 ab 2 abc b 2 c a 2 c b 2 c c 3 abc ac 2 bc 2 0 a 3 b 3 c 3 3abc 0 27. 1 c) Prove a b c 3 abc3 a 2 b 2 c 2 ab ac bca 2 b 2 c 2 ab ac bc 0a b c a 2 b 2 c 2 ab ac bc 0a 3 ab 2 ac 2 a 2b a 2 c abc a 2b b 3 bc 2 ab 2 abc b 2 c a 2 c b 2 c c 3 abc ac 2 bc 2 0 a 3 b 3 c 3 3abc 0 a b c abc1 3 3 33 28. 1 c) Prove a b c 3 abc3 a 2 b 2 c 2 ab ac bca 2 b 2 c 2 ab ac bc 0a b c a 2 b 2 c 2 ab ac bc 0a 3 ab 2 ac 2 a 2b a 2 c abc a 2b b 3 bc 2 ab 2 abc b 2 c a 2 c b 2 c c 3 abc ac 2 bc 2 0 a 3 b 3 c 3 3abc 0 a b c abc1 3 3 3311 1 let a a , b b , c c33 3 29. 1 c) Prove a b c 3 abc3 a 2 b 2 c 2 ab ac bca 2 b 2 c 2 ab ac bc 0a b c a 2 b 2 c 2 ab ac bc 0a 3 ab 2 ac 2 a 2b a 2 c abc a 2b b 3 bc 2 ab 2 abc b 2 c a 2 c b 2 c c 3 abc ac 2 bc 2 0 a 3 b 3 c 3 3abc 0 a b c abc1 3 3 3311 1 let a a , b b , c c33 3 1 1 1 1 a b c a 3 b 3 c 3 3 30. 1 c) Prove a b c 3 abc3 a 2 b 2 c 2 ab ac bca 2 b 2 c 2 ab ac bc 0a b c a 2 b 2 c 2 ab ac bc 0a 3 ab 2 ac 2 a 2b a 2 c abc a 2b b 3 bc 2 ab 2 abc b 2 c a 2 c b 2 c c 3 abc ac 2 bc 2 0 a 3 b 3 c 3 3abc 0 a b c abc1 3 3 3311 1 let a a , b b , c c33 3 1 1 1 1 a b c a 3 b 3 c 3 3 1 a b c 3 abc 3 31. Arithmetic Mean Geometric Meana1 a2 an n a1a2 ann 32. Arithmetic Mean Geometric Mean a1 a2 an n a1a2 an nd) Suppose 1 x 1 y 1 z 8, prove xyz 1 33. 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 34. 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 81 x y z 3 xyz3 35. 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 81 x y z 3 xyz3x y z 33 xyz 36. 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 81 x y z 3 xyz3x y z 33 xyzxy yz xz 33 xy yz xz 37. 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 81 x y z 3 xyz3x y z 33 xyzxy yz xz 33 xy yz xz xy yz xz 33 x 2 y 2 z 2 38. 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 81 x y z 3 xyz3x y z 33 xyzxy yz xz 33 xy yz xz xy yz xz 33 x 2 y 2 z 2xy yz xz 3 xyz 2 3 39. 1 x y z xy xz yz xyz 8 40. 1 x y z xy xz yz xyz 81 3 xyz 3 xyz xyz 8 2 3 3 41. 1 x y z xy xz yz xyz 81 3 xyz 3 xyz xyz 8 2 3 3 1 3 xyz 3 xyz xyz 8 2 33 33 42. 1 x y z xy xz yz xyz 81 3 xyz 3 xyz xyz 82 3 3 1 3 xyz 3 xyz xyz 8 2 33 3 3 1 3 xyz 83 43. 1 x y z xy xz yz xyz 81 3 xyz 3 xyz xyz 82 3 3 1 3 xyz 3 xyz xyz 8 2 33 3 3 1 3 xyz 831 3 xyz 2 44. 1 x y z xy xz yz xyz 81 3 xyz 3 xyz xyz 82 3 3 1 3 xyz 3 xyz xyz 8 233 3 3 1 3 xyz 831 3 xyz 23 xyz 1 45. 1 x y z xy xz yz xyz 81 3 xyz 3 xyz xyz 82 3 3 1 3 xyz 3 xyz xyz 8 233 3 3 1 3 xyz 831 3 xyz 23 xyz 1xyz 1 46. 1 x y z xy xz yz xyz 81 3 xyz 3 xyz xyz 82 3 3 1 3 xyz 3 xyz xyz 8 23333 1 3 xyz 831 3 xyz 23 xyz 1xyz 1 Inequalities SheetExercise 10DNote: Cambridge 8H (Book 1); 28 </p>