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. For a natural number n and any two nonnegative numbers a and b, use the Difference of Powers Formula to prove that a ≤b if and only if a" ≤b". D. For a natural number n and numbers a and b such that a ≥ b ≥ 0, prove that a"-b" ≥nb"-¹ (a - b).

. For a natural number n and any two nonnegative numbers a and b, use the Difference of Powers Formula to prove that a ≤b if and only if a" ≤b". D. For a natural number n and numbers a and b such that a ≥ b ≥ 0, prove that a"-b" ≥nb"-¹ (a - b).

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 9 and 10 please Advanced calculus I
20 ADVANCED CALCULUS Then replace b by -b to obtain |la|-|b|| ≤la - bl. 8. Let a and b be numbers such that la - b ≤ 1. Prove that a ≤ b + 1. 9. For a natural number n and any two nonnegative numbers a and b, use the Difference of Powers Formula to prove that a ≤ b if and only if a" <b". 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 n(n-1) ². (1+b)" ≥ 1+nb + 2 14. (Cauchy's Inequality) Using the fact that the square of a real number is nonnegative, prove that for any numbers a and b, neg. mult. by pos. is Sneg 2 nes is pos -2 pos is pos ab < 1 (a² + b²). 15. Use Cauchy's Inequality to prove that if a ≥ 0 and b≥ 0, then √ab ≤ ah < ≤2(a+b). 16. Use Cauchy's Inequality to show that for any numbers a and b and a natural number n,
Transcribed Image Text:20 ADVANCED CALCULUS Then replace b by -b to obtain |la|-|b|| ≤la - bl. 8. Let a and b be numbers such that la - b ≤ 1. Prove that a ≤ b + 1. 9. For a natural number n and any two nonnegative numbers a and b, use the Difference of Powers Formula to prove that a ≤ b if and only if a" <b". 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 n(n-1) ². (1+b)" ≥ 1+nb + 2 14. (Cauchy's Inequality) Using the fact that the square of a real number is nonnegative, prove that for any numbers a and b, neg. mult. by pos. is Sneg 2 nes is pos -2 pos is pos ab < 1 (a² + b²). 15. Use Cauchy's Inequality to prove that if a ≥ 0 and b≥ 0, then √ab ≤ ah < ≤2(a+b). 16. Use Cauchy's Inequality to show that for any numbers a and b and a natural number n,
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