Why do this problem?

It gives practice in working with inequalities.

As we know $n$ is a positive integer learners can investigate $n^{{1\over n}}$ for for different values of $n$ and make conjectures about where the maximum value occurs.

Possible approach

You need to find a local maximum for a small value of $n$ and then prove that this is the only maximum value. Clearly it is impossible to check all values of $n$. One method of proving the result uses the Binomial theorem.