#### Question

Show that f(x) = e^{1}^{/x}, x ≠ 0 is a decreasing function for all x ≠ 0 ?

#### Solution

\[f\left( x \right) = e^\frac{1}{x} \]

\[f'\left( x \right) = e^\frac{1}{x} \frac{d}{dx}\left( \frac{1}{x} \right)\]

\[ = e^\frac{1}{x} \left( \frac{- 1}{x^2} \right)\]

\[ = - \frac{e^\frac{1}{x}}{x^2}\]

\[\text { Here, }e^\frac{1}{x} > 0 \text { and } x^2 > 0, \text { for any real value of} x \neq 0.\]

\[\therefore f \left( x \right) = - \frac{e^\frac{1}{x}}{x^2} < 0, \forall x \in R, x \neq 0\]

\[\text { So,f(x) is a decreasing function }.\]

Is there an error in this question or solution?

Solution Show that F(X) = E1/X, X ≠ 0 is a Decreasing Function for All X ≠ 0 ? Concept: Increasing and Decreasing Functions.