Question: (a) Prove that \(x+\frac{1}{x y}+y^{2} \geq 5\left(2^{-\frac{4}{5}}\right)\) for \(x, y>0\) and determine the values of \(x\) and \(y\) which make this an equality.

(b) Given that \(a_{i j}>0(i=1, \ldots, m ; j=1, \ldots, n)\) and that \(\alpha_{1}, \ldots, \alpha_{m}>0\)

with \(\alpha_{1}+\ldots+\alpha_{m}=1\), prove that \(\left(a_{11}^{\alpha_{1}} \ldots a_{m 1}^{\alpha_{m}}\right)+\ldots+\left(a_{1 n}^{\alpha_{1}} \ldots a_{m n}^{\alpha_{m}}\right) \leq\)

\(\left(a_{11}+\ldots+a_{1 n}\right)^{\alpha_{1}} \ldots\left(a_{m 1}+\ldots+a_{m n}\right)^{\alpha_{m}}\)

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