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Mathematics · 2026 · 2 marks
CBSE 2026 · Region 4 · Set 2 · Q23
Find the interval(s) in which the function f ( x )=x/(log x), where x ∈(0,1) ∪(1, ∞), is increasing.
Find the interval(s) in which the function $\displaystyle \mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{\log \mathrm{x}}$, where $\displaystyle \mathrm{x} \in(0,1) \cup(1, \infty)$, is increasing.
Marking-scheme solution
$\displaystyle f^{\prime}(x)=\dfrac{\log x-1}{(\log x)^{2}}$
Put $\displaystyle f^{\prime}(x)=0 \Rightarrow \log x=1$
As, $\displaystyle \log x=1=\log e \Rightarrow x=e$
$\displaystyle \therefore f(x)$ is increasing on $\displaystyle x \in(e, \infty)$ or $\displaystyle x \in[e, \infty)$.
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