e^pi vs. pi^e

Date: 03/20/2002 at 14:51:19
From: Rajesh Gunesh
Subject: Number Theory

Which is greater, e^pi or pi^e? I would like to have a simple proof,
please. 

Thank you!

Date: 03/20/2002 at 15:28:22
From: Doctor Paul
Subject: Re: Number Theory

Consider the function f(x) = e^x - x^e. Its derivative is 

f'(x) = e^x - e*x^(e-1)
f'(x) = 0 will have at most two solutions since f'(x) = 0 implies 

e^x = e*x^(e-1)

and it is easily seen that an exponential function and a power 
function have at most two points of intersection. Observe that 
f'(1) = 0 and f'(e) = 0, so x = 1 and x = e are the two points of 
intersection. 

In general, an exponential function will grow to infinity faster than 
a power function, so f'(x) >= 0 for x > e. This means that f is an 
increasing function when x > e.  Also notice that f(e) = 0. Thus, 
since Pi > e and f is increasing when x > e, we have f(Pi) > 0.

Thus f(Pi) = e^Pi - Pi^e > 0. This implies that e^Pi is greater than 
Pi^e. 

A calculator verifies that e^Pi ~= 23.14 and Pi^e ~= 22.46

I hope this helps.  Please write back if you'd like to talk about this 
some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/