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sing the fact that the square bers a and b. ab ≤ (a² + b²). to prove that if a ≥ 0 and

sing the fact that the square bers a and b. ab ≤ (a² + b²). to prove that if a ≥ 0 and

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve number 14
Then rep 9. For a natural number n and any two nonnegative numbers a and b, use the Difference 8. Let a and b be numbers such that la- b ≤ 1. Prove that a ≤ b+1. of Powers Formula to prove that a ≤ b if and only if a ≤ bn. 10. For a natural number n and numbers a and b such that a ≥ b ≥ 0, prove that a"-b" ≥nb"-1(a - b). 11. (Bernoulli's Inequality) Show that for a natural number n and a nonnegative number b, (1+b)" ≥ 1+nb. (Hint: In the Binomial Formula, set a = 1.) 12. Use the Principle of Mathematical Induction to provide a direct proof of Bernoulli's Inequality for all b> -1, not just for the case where b ≥ 0 which, as outlined in Exercise 11 follows from the Binomial Formula. 13. For a natural number n and a nonnegative number b show that (1+b)" ≥ 1+nb + 14. (Cauchy's Inequality) Using the fact that the square of a real number is nonnegative, prove that for any numbers a and b, is neg. mult. by pos. оред -2 nes is pos -2 pos is pos ab ≤ 1 n(n-1) ². 2 ab < (a² + b²). 15. Use Cauchy's Inequality to prove that if a ≥ 0 and b≥ 0, then 1 √ab ≤ = (a + b). 16. Use Cauchy's Inequality to show that for any numbers a and b and a natural number n, - na² + (Hint: Replace a by √√na and b by b/√√n in Cauchy's Inequality) 17. Let a, b, and c be nonnegative numh a. ab + bc + ca <a² n
Transcribed Image Text:Then rep 9. For a natural number n and any two nonnegative numbers a and b, use the Difference 8. Let a and b be numbers such that la- b ≤ 1. Prove that a ≤ b+1. of Powers Formula to prove that a ≤ b if and only if a ≤ bn. 10. For a natural number n and numbers a and b such that a ≥ b ≥ 0, prove that a"-b" ≥nb"-1(a - b). 11. (Bernoulli's Inequality) Show that for a natural number n and a nonnegative number b, (1+b)" ≥ 1+nb. (Hint: In the Binomial Formula, set a = 1.) 12. Use the Principle of Mathematical Induction to provide a direct proof of Bernoulli's Inequality for all b> -1, not just for the case where b ≥ 0 which, as outlined in Exercise 11 follows from the Binomial Formula. 13. For a natural number n and a nonnegative number b show that (1+b)" ≥ 1+nb + 14. (Cauchy's Inequality) Using the fact that the square of a real number is nonnegative, prove that for any numbers a and b, is neg. mult. by pos. оред -2 nes is pos -2 pos is pos ab ≤ 1 n(n-1) ². 2 ab < (a² + b²). 15. Use Cauchy's Inequality to prove that if a ≥ 0 and b≥ 0, then 1 √ab ≤ = (a + b). 16. Use Cauchy's Inequality to show that for any numbers a and b and a natural number n, - na² + (Hint: Replace a by √√na and b by b/√√n in Cauchy's Inequality) 17. Let a, b, and c be nonnegative numh a. ab + bc + ca <a² n
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