Exponents Homework: Proving Inequality of Positive Integers

[furnis1]

Homework Statement

Hey guys, I am having difficulty with the following problem:

"If m and n are two postive integers, prove that one of m^(1/n) or n^(1/m) is always less than or equal to 3^(1/3)"

Any idea of how to go about this?

Homework Equations

The Attempt at a Solution

 

[futurebird]

Hmm...

Well here is what could happen:

m=n
m>n or
m<n

The last two cases can be treated as one.

One more hint: for what value of x is x^{\frac{1}{x}} maximized? I think it's e.

 

[dirk_mec1]

Hmm...
One more hint: for what value of x is x^{\frac{1}{x}} maximized? I think it's e.

I don't see how e can be useful since 3^{1/3} \leq e

 

[futurebird]

If you know where the max value is you should be able to locate the max value for the function on the positive integers by looking at where the function is increasing and decreasing.

Then deal with the case where m != n