find the largest no among these

Question

$1,2^{1/2},3^{1/3},4^{1/4},\dotsc $

Welcome to Math.SE ! Please share your efforts in solving this problem otherwise it is likely to get closed — Shailesh, Aug 08 '16 at 15:59

score 3 · Answer 1 · answered Aug 08 '16 at 15:55

3

Hint: the function $f(x)=x^{1/x}$ has a maximum point at $e$ in $(0,+\infty)$. Consider the derivative $$f'(x)=\frac{x^{1/x}(1-\ln(x))}{x^2}.$$

answered Aug 08 '16 at 15:55

Robert Z

Here not f(x)....it is f(n)=n^(1÷n) – sasanka jana Aug 08 '16 at 16:46
@sasanka jana Yes, but the derivative says that $f$ is increasing in $(0,e)$ and decreasing in $(e,+\infty)$. So $f(1)<f(2)$ and $f(3)>f(4)>f(5)>\cdots$. So the maximum is $f(2)=2^{1/2}$ or $f(3)=3^{1/3}$. Raising to the 6th power you find easily that $8<9$ implies that $2 ^{1/2} < 3^{1/3}$. By the way, I gave only a hint in my answer... – Robert Z Aug 08 '16 at 17:03
Again lim n^{1/n}=1 as n tends to infinity – sasanka jana Aug 08 '16 at 18:14

score 3 · Answer 2 · answered Aug 08 '16 at 16:01

3

Since $x^{1/x}$ has a unique maximum at $x=e$, the only possibilities are $2$ and $3$.

Since $2^3 < 3^2$, $2^{1/2} < 3^{1/3}$, the maximum is at $x=3$.

answered Aug 08 '16 at 16:01

marty cohen

2 Answers2