For a>=b>=c>0. Prove that a/(a+b)+b/(b+c)+c/(c+a)>=3/2 a=100;b=1;c=1/100 it's wrong ???

djo57bgfrqn

djo57bgfrqn

Answered question

2022-10-15

Prove that a a + b + b b + c + c c + a 3 2
For a b c > 0. Prove that
a = 100 ; b = 1 ; c = 1 / 100 it's wrong ???
a a + b + b b + c + c c + a 3 2
<=> a a + b 1 2 + b b + c 1 2 + c c + a 1 2 0
<=> a b 2 ( a + b ) 0
<=> x y 2 + y z 2 + x z 2 x 2 y y 2 z z 2 x 0
<=> ( y x ) ( z y ) ( z x ) 0
And we have y x 0 ; z x 0 ; z y 0
we are done !?

Answer & Explanation

jbaekk7q

jbaekk7q

Beginner2022-10-16Added 18 answers

For a b c > 0 we obtain:
c y c a a + b 3 2 = c y c ( a a + b 1 2 ) = c y c a b 2 ( a + b ) =
= c y c ( a b ) ( c 2 + a b + a c + b c ) 2 c y c ( a + b ) = c y c c 2 a c 2 b 2 c y c ( a + b ) =
= c y c a 2 b a 2 c 2 c y c ( a + b ) = ( a b ) ( a c ) ( b c ) 2 c y c ( a + b ) 0

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?