Lesson1. AM−GM ′ s inequality (Cauchy)
form : ((a_1 +a_2 +...+a_n )/n) ≥ ((a_1 a_2 ...a_n ))^(1/n)
where a_1 ,a_2 ,....,a_n >0
Equal at a_1 =a_2 =.....=a_n
e.g. 1. Given a,b,c>0, prove that
(a+b)(b+c)(c+a)≥8abc
Solu. by AM−GM
a+b ≥ 2(√(ab)) (1)
b+c ≥ 2(√(bc)) (2)
c+a ≥ 2(√(ca)) (3)
(1)×(2)×(3) ⇒ (a+b)(b+c)(c+a)≥8(√(a^2 b^2 c^2 ))=8abc
Now practice.
. Given a,b,c>0 prove that
1. a^2 +b^2 +c^2 ≥ab+bc+ca
2. (a+(1/b))(b+(1/c))(c+(1/a))≥8
3. 4(a^3 +b^3 )≥(a+b)^3
4. 9(a^3 +b^3 +c^3 )≥(a+b+c)^3
let′s try, I will post my solution for which one
that you can′t do ;)
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