I have a book, which states this inequality and need to be proof. Absolutely, this is a problem.

Given \(a,b \in \mathbb{R}^+\) and \(a\neq b\), proof that \[a^{3} +b^{3} > a^{2}b +ab^{2}\]

Proof: since \(a \neq b\), obviously that \[a+b>0\] \[(a-b)^2>0\]

Multiply both ineqs and we have \[(a+b)(a-b)^2>0\] \[(a+b)(a^2-2ab+b^2)>0\] \[a^3-a^2b-ab^2+b^3>0\] \[a^3+b^3>a^2b + ab^2\]

Done!

Is that valid? Comments will be appreciated!

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