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Mathematics · 2023 · 2 marks
CBSE 2023 · Region 3 · Set 2 · Q21
Consider the statement "There exists at least one value of b ∈ R for which f ( x )=b/x, b ≠ 0 is strictly increasing in R -{0}." State True or False.…
Consider the statement "There exists at least one value of $\displaystyle \mathrm{b} \in \mathbb{R}$ for which $\displaystyle \mathrm{f}(\mathrm{x})=\frac{\mathrm{b}}{\mathrm{x}}, \mathrm{b} \neq 0$ is strictly increasing in $\displaystyle \mathbb{R}-\{0\}$." State True or False. Justify.
Marking-scheme solution
The given statement is "True".
$$\mathrm{f}^{\prime}(\mathrm{x})=-\frac{\mathrm{b}}{\mathrm{x}^{2}}
$$
for $\displaystyle \mathrm{b}<0, \mathrm{f}^{\prime}(\mathrm{x})>0$ in $\displaystyle (-\infty, 0)$ and $\displaystyle (0, \infty)$
$\displaystyle \therefore \mathrm{f}(\mathrm{x})$ is strictly increasing in both these intervals.
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