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Question
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Solution
\[f\left( x \right) = e^\frac{1}{x} \]
\[f'\left( x \right) = e^\frac{1}{x} \frac{d}{dx}\left( \frac{1}{x} \right)\]
\[ = e^\frac{1}{x} \left( \frac{- 1}{x^2} \right)\]
\[ = - \frac{e^\frac{1}{x}}{x^2}\]
\[\text { Here, }e^\frac{1}{x} > 0 \text { and } x^2 > 0, \text { for any real value of} x \neq 0.\]
\[\therefore f \left( x \right) = - \frac{e^\frac{1}{x}}{x^2} < 0, \forall x \in R, x \neq 0\]
\[\text { So,f(x) is a decreasing function }.\]
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