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Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives1 Mark
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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