$\displaystyle \ 1,\sqrt{2},\sqrt[3]{3}.....,\sqrt[n]{n}$
Determine the largest number in the above sequence
The function $\displaystyle y = x^{1/x}$ has a maximum turning point at x = e. Since 2 < e < 3, the question now becomes a simpler one:
"Which is bigger, $\displaystyle \sqrt{2}$ or $\displaystyle \sqrt[3]{3}$ " .... Which is more or less the same as asking "Which is bigger, $\displaystyle (\sqrt{2})^6$ or $\displaystyle (\sqrt[3]{3})^6$ " ....
Alternatively, note that $\displaystyle \sqrt{2}$ has the same value as $\displaystyle \sqrt[4]{4} = 4^{1/4}$ .....