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Question
Find the intervals of monotonicities and hence find the local extremum for the following functions:
f(x) = `"e"^x/(1 - "e"^x)`
Solution
f'(x) = `((1 - "e"^x)"e"^x - "e"^x(- "e"^x))/(1 - "e"^x)^2`
= `("e"^x - "e"^(2x) + "e"^(2x))/(1 - "e"^x)^2`
= `"e"^x/(1 - "e"^x)^2`
When x = 0, f(x) becomes undefined.
∴ x = 0 is an excluded value.
∴ The intervals are `(-oo, 0)` ∪ `(0, oo)` in `– (-oo, oo)`, f'(x) > 0
∴ f(x) is strictly increasing in `(- oo, oo)` and there is no extremum.
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