Hello all of you! If you are somehow bored with B/W and you really have nothing better to do, here is a problem that I read about:

Which is greater: e^pi or pi^e ?

This should not be too difficult for everyone with a reasonable foundation in calculus. It is not designed to be obvious. Please try to solve this without using a calculator! For those of you who are stumped, I have enclosed a hint.

Hint:

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

e^x>x^e, always.

Well, at least for real nonnegative values of x.

Oh right.