Show that for \(x>0\), \(x^{1/x}\) has its greatest value when \(x=e\).

Hence, or otherwise, but without use of a calculator, determine which is the greater of \(e^{\pi}\) and \(\pi^e\).

Show that for \(x>0\), \(x^{1/x}\) has its greatest value when \(x=e\).

Hence, or otherwise, but without use of a calculator, determine which is the greater of \(e^{\pi}\) and \(\pi^e\).