Question

If $0<x<1$, what is the value of $x+\frac{1}{x}$?

• A. Greater than $2$
• B. Less than $2$
• C. Greater than $4$
• D. Less than $4$

Answer 1

The correct option is A Greater than $2$

$f\left(x\right)=x+\frac{1}{x}$

$\underset{x\to 0}{Lim}f\left(x\right)=\infty$

and $f^{{}^{\prime }}\left(x\right)=1-\frac{1}{x^2}$

$f^{{}^{\prime \prime }}\left(x\right)=1+\frac{2}{x^3}$

for $xϵ(0,1);$$f^{{}^{\prime \prime }}\left(x\right)>0$

so, $f\left(x\right)$ is decreasing in $xϵ(0,1)$

so minima occurs at $\underset{x\to 1}{Lim}f\left(x\right)$

and $\underset{x\to 1}{Lim}f\left(x\right)=2$, so $f\left(x\right)$ is greater then $2$