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…

This looks tricky as \(z\) is given as a function of two variables. Can we use the equation connecting \(x\) and \(y\) to turn \(z\) into a function of just one variable?

How do we then calculate stationary points for \(z\)?

… and show that this value of \(z\) is a minimum.

What does the second derivative tell us about a stationary point?