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High powered Inequality

Prove \[x^4+y^4+z^4\geq xyz(\sqrt{xy}+\sqrt{yz}+\sqrt{xz})\].

Note by Joshua Ong
3 years, 6 months ago

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Use AM-GM.

Joshua Ong - 3 years, 6 months ago

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The result follows immediately from Muirhead: \((4,0,0)\succ \left(\dfrac{3}{2},\dfrac{3}{2},1\right)\)

Daniel Liu - 3 years, 6 months ago

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Except it is not given that \(x,y,z>0\) and Muirhead's inequality holds for positive reals.

Mathh Mathh - 3 years, 6 months ago

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Are \(x,y,z\) positive reals or just reals? (the same question for some of the other problems. You should provide the given restrictions on \(x,y,z\) since otherwise we're not quite sure what they are).

Mathh Mathh - 3 years, 6 months ago

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