For positive real numbers a , b prove that <mroot> 2 ( a +

Saniya Copeland

Saniya Copeland

Answered question

2022-06-02

For positive real numbers a , b prove that
2 ( a + b ) ( 1 a + 1 b ) 3 a b 3 + b a 3

Answer & Explanation

tabustudiofx52n

tabustudiofx52n

Beginner2022-06-03Added 6 answers

Starting from Aqua answer, with x = b a 3 , but then let y = ( x + 1 x )
y 3 = ( x + 1 x ) 3 = ( x 3 + 1 x 3 ) + 3 y
We have to proof:
4 + 2 · ( y 3 3 y ) y 3 y 3 6 y + 4 0
Using AM-GM inequality, y 2
Let z = y 2
( z + 2 ) 3 6 ( z + 2 ) + 4 = ( z 3 + 6 z 2 + 12 z + 8 ) 6 z 8 = z 3 + 6 z 2 + 6 z 0
With positive coefficients, and z 0, above is always true. QED

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