\[ e^\pi> \pi^e.\]
How would we go about convincing someone that this amazing inequality is true?
There are several ways to get a feel for this inequality, here are some suggestions:
- what do the functions \(e^x\) and \(x^e\) look like?
- how could we approximate \(e\) and \(\pi\)?
- do we know any functions that change a question about indices into a question about something else?
How do we define \(e^\pi\) and \(\pi^e\) given that both numbers are transcendental?