The variables \(x\) and \(y\) are such that \(x^4y=8\). A third variable \(z\) is defined by \(z=x+y\).

Find the values of \(x\) and \(y\) that give \(z\) a stationary value and show that this value of \(z\) is a minimum.

The variables \(x\) and \(y\) are such that \(x^4y=8\). A third variable \(z\) is defined by \(z=x+y\).

Find the values of \(x\) and \(y\) that give \(z\) a stationary value and show that this value of \(z\) is a minimum.