Determining the bigger number

1. Sep 17, 2013

Saitama

1. The problem statement, all variables and given/known data
Use the function $f(x)=x^{1/x} \, ,\, x>0$, to determine the bigger of two numbers $e^{\pi}$ and $\pi^e$.

2. Relevant equations

3. The attempt at a solution
I honestly don't know where to begin with this problem. I found the derivative but that seems to be of no help. The function increases when x>1/e and decreases when 0<x<1/e.

Any help is appreciated. Thanks!

2. Sep 17, 2013

Staff: Mentor

Hint: a<b is equivalent to a^c < b^c for positive a,b,c. Can you find a c such that your expressions look like x^(1/x)?

3. Sep 17, 2013

haruspex

Doesn't sound right. What do you get for x = 1, 2, 4, 16?

4. Sep 17, 2013

Saitama

Oops, its just the opposite of what I posted.

I have tried the problem again.
$f(e)>f(\pi) \Rightarrow e^{1/e}>\pi^{1/\pi}$
Raising both the sides to the power $e\pi$
$$e^{\pi}>\pi^e$$

Looks correct?

5. Sep 17, 2013

Dick

Looks correct.

6. Sep 17, 2013

Saitama

Thank you Dick and haruspex!