Let a, b, c are positive numbers. Prove

Eddie Clarke

Eddie Clarke

Answered question

2022-04-14

Let a, b, c are positive numbers. Prove that a2(a+b)24a2b2bc+7ca4(a+b+c)2

Answer & Explanation

ditumpasz9xj

ditumpasz9xj

Beginner2022-04-15Added 8 answers

We need to prove that:
4a2c2+(a+b)(a26ab+b2)c+b2(a+b)20,
which is a quadratic inequality of c.
If a26ab+b20, it's obviously true.
But for a26ab+b20 it's enough to prove that
(a26ab+b2)216a2b20
or (a22ab+b2)(a210ab+b2)0,
which is obvious.

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