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Question
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
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Solution
\[\text{ Here }, \]
\[f\left( x \right) = \frac{1}{x}\]
\[\text { Let } x_1 , x_2 \in \left( 0, \infty \right) \text { such that } x_1 < x_2 . \text { Then }, \]
\[ x_1 < x_2 \]
\[ \Rightarrow \frac{1}{x_1} > \frac{1}{x_2}\]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So, }f\left( x \right)\text { is decreasing on }\left( 0, \infty \right) .\]
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