Give answers with COMPLETED proof steps
Posted: Tue Sep 07, 2021 7:38 am
Give answers with COMPLETED proof steps
1. Va, x ER if ax=1 then -1 x=9-1. Here a is the multiplicative inverse of a 2. For all nen, n>0 3. V a,b ERifO<a<b then 1 1 0< —<- a Here 1 and 1 are the multiplicative inverses of b and a. b a
1.2.0 Let a, b, c, d e R and consider each of the following statements. Decide which are true and which are false. Prove the true ones and give coun- terexamples to the false ones. b) If a <b and c > 1, then la + c = 1b+cl. c) If a sband b Sa+c, then la - bisc.
Let a, b ER a) Prove that if a > 2 and b=l+Va- 1, then 2 <b<a.
The arithmetic mean of a, b e R is A(a, b) = (a+b)/2, and the geometric mean of a, b € [0,00) is G(a, b) = Vab. If 0 <a <b, prove that a < G(a, b) < A(a,b) <b. Prove that G(a,b) = A(a, b) if and only if a=b.
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1.2.0 Let a, b, c, d e R and consider each of the following statements. Decide which are true and which are false. Prove the true ones and give coun- terexamples to the false ones. b) If a <b and c > 1, then la + c = 1b+cl. c) If a sband b Sa+c, then la - bisc.
Let a, b ER a) Prove that if a > 2 and b=l+Va- 1, then 2 <b<a.
The arithmetic mean of a, b e R is A(a, b) = (a+b)/2, and the geometric mean of a, b € [0,00) is G(a, b) = Vab. If 0 <a <b, prove that a < G(a, b) < A(a,b) <b. Prove that G(a,b) = A(a, b) if and only if a=b.