e^pi or pi^e is greater.

Lets consider e^x and x^e
When x is 0, e^x = 1 and x^e is 0
Now we know e^x > x^e when x is 0

Lets find the mininum difference between the two
Now consider the equation
y = e^x - x^e
When the value of y is minimum, dy/dx = 0
=> d/dx (e^x - x^e) = 0
=> e^x - e x^(e-1) = 0

=> e^x - e(x^e)/x = 0
=> x(e^x) - e(x^e) = 0
=> x = e

Hence the minimum value of y is at x = e
The value of y at x=e is 0

Now that we proved e^x - x^e is positive at x = 0 and the minimum is 0 at x = e,
this implies e^x - x^e is positive for all the values other than x = e

=> e^x - x^e > 0
=> e^x > x^e

Hence for any value of x including pi, e^x > x^e

Therefore e^pi > pi^e