For a,b,c are positive real number satisfy a+b+c=3. Prove that (a^2)/(a+root[3]{bc))+(b^2)/(b+root[3](ca))+(c^2)/(c+root[3](ab))>=(3)/(2)

Plaginicj

Plaginicj

Answered question

2022-09-04

For a , b , c. Prove that a 2 a + b c 3 + b 2 b + c a 3 + c 2 c + a b 3 3 2
By Cauchy-Schwarz: a 2 a + b c 3 + b 2 b + c a 3 + c 2 c + a b 3 ( a + b + c ) 2 a + b + c + b c 3 + c a 3 + a b 3
Hence we need prove 9 3 + b c 3 + c a 3 + a b 3 3 2 ( )
9 3 a b 3 + 3 b c 3 + 3 c a 3
By AM-GM a + b + 1 3 a b 3 and . . .
2 ( a + b + c ) + 3 3 a b 3 + 3 b c 3 + 3 c a 3
9 3 a b 3 + 3 b c 3 + 3 c a 3
I need a new method.

Answer & Explanation

Belen Solomon

Belen Solomon

Beginner2022-09-05Added 5 answers

By AM-GM and C-S we obtain:
c y c a 2 a + b c 3 c y c a 2 a + b + c + 1 3 = c y c 9 a 2 3 ( 3 a + b + c ) + a + b + c =
= c y c 9 a 2 10 a + 4 b + 4 c 9 ( a + b + c ) 2 c y c ( 10 a + 4 b + 4 c ) = 9 ( a + b + c ) 2 18 ( a + b + c ) = 3 2

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