Question

Consider the set $P=\left\{n^{\frac{1}{n}} , n \in N\right\},$ then the largest element of set $P$ is

• A. $2^{\frac{1}{2}}$
• B. $3^{\frac{1}{3}}$
• C. $e^{\frac{1}{e}}$
• D. $7^{\frac{1}{7}}$

Answer 1

Answer: (B)

Let $f \left(x\right) = x^{\frac{1}{x}} \Rightarrow f^{'} \left(x\right) = x^{\frac{1}{x}} \left(\frac{1 - \text{ln} x}{x^{2}}\right)$
At $x = e , f^{'} \left(x\right) = 0$ and sign change of derivative is from positive to negative, hence $x=e$ is a point of local maxima of $f\left(x\right).$
Now $2 < e < 3$ , hence $f\left(2\right)$ or $f\left(3\right)$ is the greatest element of set $P$
Since $2^{3} < 3^{2}\Rightarrow 2^{\frac{1}{2}} < 3^{\frac{1}{3}}$
So, $f\left(3\right)=3^{\frac{1}{3}}$ is the largest element of set $P$