Show that the only stationary point on the curve \(\mathrm { y } = \frac { \ln \mathrm { x } } { \mathrm { x } }\) occurs where \(x = \mathrm { e }\), as given in line 45.
Show that the stationary point is a maximum.
It follows from part (b) that, for any positive number \(a\) with \(a \neq \mathrm { e }\),
\(\frac { \ln \mathrm { e } } { \mathrm { e } } > \frac { \ln a } { a }\).
Use this fact to show that \(\mathrm { e } ^ { a } > a ^ { \mathrm { e } }\).