As in the title. Prove that for nonnegative $a$ and $b$ the following inequality holds: $$(a+b)^4\ge8ab(a^2+b^2).$$ Note that I'm not looking for a complete solution, but only for some hints.
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Hint: $$(a+b)^4-(a-b)^4 = 8ab(a^2+b^2)$$
Thomas Andrews
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And... now it's completely trivial. Great. Thanks. – Feb 14 '16 at 08:37
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A hint could be this one
$$(a + b)^4 = (a+b)^2(a+b)^2 = (a^2 + b^2)(a+b)^2 + 2ab(a^2 + b^2) + 4a^2b^2$$
Or similar manipulations!
Enrico M.
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5the hint doesn't help as the $RHS \geq 4ab(a^2+b^2) + 2ab(a^2+b^2) + 4a^2b^2 = 6ab(a^2+b^2) + 4a^2b^2$ – DeepSea Feb 14 '16 at 00:07