The function f(x)=x x increases in the interval - Vidyadip

MCQ

The function $f(x)=\frac{x}{\log x}$ increases in the interval

A
$(0, \infty)$
B
$(0, e)$
✓
$(e, \infty)$
D
None of these

✓

Answer

Correct option: C.

$(e, \infty)$

(c) : $f(x)=\frac{x}{\log x}$ is defined for $x>0$ and $x \neq 1$
Also, $f^{\prime}(x)=\frac{\log x \cdot 1-x \cdot \frac{1}{x}}{(\log x)^2}=\frac{\log x-1}{(\log x)^2}$
$
\therefore f^{\prime}(x)>0 \Rightarrow \log x>1 \Rightarrow x>e \therefore x \in(e, \infty) \text {.}
$

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