Question

The function $y = \frac{x}{\log x}$ increases in the interval

• A. $(e,\infty)$
• B. $\lbrack e,\infty)$
• C. $(0,e)$
• D. $\lbrack 0,e\rbrack$

Answer 1

Answer: (B)

$\lbrack e,\infty)$

Answer 2

$y' = \frac{\log x - 1}{\left( \log x \right)^{2}}$

Since $\left( \log x \right)^{2}$is always positive, we have $y' > 0$ if $\log x > 1$. That is, if $x > e,$ or if $x \in (e,\infty)$ Therefore, $y$ increases on $(e,\infty)$ or any interval contained in $(e,\infty)$.