Hi Prof Lee,
Thanks for the lunch (including the advice part). I came up with some ideas about this brain teaser —
Q: which is bigger e^pi vs pi^e
One solution I can think of is, suppose e has a value close to 2 and pi is much larger.
Suppose e = 2 and pi = 10. Clearly e^pi wins.
Another way is, define 2 functions
f1(x) = 2.718281828^x and find the growth rate when x is slightly above e. This growth rate is e^x,
f2(x) = x^2.718281828 and find the growth rate when x is slightly above e. This grow rate is e/x * x^e, which is smaller, since x is slightly bigger than e.
Therefore, f1 grows faster than f2, over the range of (e , 3.15). Therefore e^pi wins.