SOLUTION: if a and b are two distinct non-negative numbers, prove that a^4+b^4>ab(a^2+b^2)

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Question 461590: if a and b are two distinct non-negative numbers, prove that a^4+b^4>ab(a^2+b^2)
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Not both of a and b are equal to zero, by the given. Assume one of them, say a, is zero, but b+%3C%3E+0. Then b%5E4+%3E+0%2Ab%2A%280%2Bb%29+=+0, which is always true.
Hence assume that both a and b are positive.
Then
%28a-b%29%5E2%28a%5E2+%2B+ab+%2B+b%5E2%29+%3E+0
<==> %28a-b%29%28a-b%29%28a%5E2+%2B+ab+%2B+b%5E2%29+%3E+0
<==> %28a%5E3-b%5E3%29%28a+-+b%29+%3E+0
<==> a%5E3%28a-b%29+-+b%5E3%28a-b%29+%3E+0
<==> a%5E3%28a-b%29+%2B+b%5E3%28b-a%29+%3E+0+
<==> a%5E4+-+a%5E3b+%2B+b%5E4+-+ab%5E3+%3E+0
<==> a%5E4+%2B+b%5E4+%3E+a%5E3b+%2B+ab%5E3
<==> a%5E4+%2B+b%5E4+%3E+ab%28a%5E2+%2B+b%5E2%29