Comparing Numbers Using a Special Function

[Saitama]

Homework Statement

Use the function $f(x)=x^{1/x} \, ,\, x>0$, to determine the bigger of two numbers $e^{\pi}$ and $\pi^e$.

Homework Equations

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!

 

[mfb]

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)?

 

[haruspex]

. The function increases when x>1/e and decreases when 0<x<1/e.

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

 

[Saitama]

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

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?

 

[Dick]

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?

Looks correct.

 

[Saitama]

Looks correct.

Thank you Dick and haruspex!