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Mathematics · 2026 · 2 marks
CBSE 2026 · Region 3 · Set 3 · Q23
Find the sub-interval of (0, ∞) in which f ( x )= x log x is increasing.
Find the sub-interval of $\displaystyle (0, \infty)$ in which $\displaystyle \mathrm{f}(\mathrm{x})=\mathrm{x} \log \mathrm{x}$ is increasing.
Marking-scheme solution
$\displaystyle \mathrm{f}(x)=x \log x \Rightarrow \mathrm{f}^{\prime}(x)=1+\log x$
Put $\displaystyle 1+\log x=0 \Rightarrow x=\dfrac{1}{e}$
$\displaystyle \because \mathrm{f}^{\prime}(x)>0\ \forall x \in\left(\dfrac{1}{e}, \infty\right) \Rightarrow \mathrm{f}(x)$ is increasing in $\displaystyle \left(\dfrac{1}{e}, \infty\right)$.
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