Good Question!

Any hints ? :P

hint : consider the function x^(1/x) ..analyse where it is increasing ...decreasing ..its maxima ..etc etc

qwerty's hint should finish off the problem

consider f(x) = x^1/x. Verify that f_max = f(e) = e^1/e
hence e^1/e>pi^1/pi implying e^pi> pi^e

let y = x^1/x .

dy/dx = x^{(1/x - 2)} + x^1/x*{lnx}(-1/x²)
which is < 0 for x > 1

Hence y is decreasing for x > 1

=> y(pi) < y(e)

(pi)^1/pi < e^1/e

raise both sides to the power e*pi

=> pi^e < e^pi

Shubodip solution is right

@Rishabh
f'(x)=0
when ln(x)=1
i.e
at x=e

so f(max)=f(e)

thanks for the correction...silly me