Question

The term with the maximum value among the terms of the sequence $1,2^{\frac{1}{2}},3^{\frac{1}{3}},4^{\frac{1}{4}},....$ is

• A. the first term
• B. the second term
• C. the third term
• D. the fifth term

Answer 1

The correct option is C the third term

Let $f\left(x\right)=y=x^{\frac{1}{x}}$
$\Rightarrow \ln y=\frac{1}{x}\ln x\Rightarrow x\ln y=\ln x$
Differentiating wrt $x,$ we get
$\ln y+\frac{x}{y}{y}^{\prime }=\frac{1}{x}$

For maxima, $y^{\prime }=0$
$\Rightarrow \ln y=\frac{1}{x}\Rightarrow \frac{1}{x}\ln x=\frac{1}{x}$
$\Rightarrow \ln x=1\Rightarrow x=e$
We can infer that $x=e$ is the point of maxima.

So, $f\left(x\right)$ is maximum either at $x=2$ or $x=3.$
$f\left(2\right)=2^{\frac{1}{2}}=1.414$
$f\left(3\right)=3^{\frac{1}{3}}=1.44$
So, the maximum value is $3^{\frac{1}{3}}$ which is the third term.