Question

whats bigger
e^pi or pi^e?
do not use calculator and give an explanation for your ans.

Answer 1

hint: it can be solved using calculus

Answer 2

\[(2.7)^{3.14}\]

Or,

\[(3.14)^{2.7}\]

Answer 3

the results are too close for a numerical approach

Answer 4

both e^pi and pi^e
are positive
now raise both to the power
-e*pi
which makes them
e^1/e and pi^1/pi

Answer 5

I think pi^e is greater...

Answer 6

now they are of the form x^1/x
hence if u diff
we get \[[x ^{1/x}/x ^{2}] * [1-\ln x]\]

Answer 7

which means f'x is negative for x >e
hence as x increases after x=e
f(x) keeps decreasing
meaning
pi^e < e^pi

Answer 8

Here's one way:

Consider the function f(x) = x / ln(x). Its derivative is
f ' (x) = 1*(1/ln(x)) + x[ -(ln(x))^-2 ]*(1/x)
f ' (x) = (1/ln(x)) - 1/(ln(x))^2
f ' (x) = (ln(x) - 1) / [ln(x)]^2
Setting this equal to 0 shows that it reaches a local minimum when x=e. So since e < pi, that must mean f(e) < f(pi). This means:

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

So e^pi is greater.

Answer 9

lovely man

Answer 10

hey nitz how did u think of it?