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Question
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
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Solution
\[f\left( x \right) = \left( x + 2 \right) e^{- x} \]
\[f'\left( x \right) = - e^{- x} \left( x + 2 \right) + e^{- x} \]
\[ = - x e^{- x} - 2 e^{- x} + e^{- x} \]
\[ = - x e^{- x} - e^{- x} \]
\[ = e^{- x} \left( - x - 1 \right)\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow e^{- x} \left( - x - 1 \right) > 0\]
\[ \Rightarrow - x - 1 > 0 \left[ \because e^{- x} > 0, \forall x \in R \right]\]
\[ \Rightarrow - x > 1\]
\[ \Rightarrow x < - 1\]
\[ \Rightarrow x \in \left( - \infty , - 1 \right)\]
\[\text { So, f(x) is increasing on} \left( - \infty , - 1 \right) . \]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow e^{- x} \left( - x - 1 \right) < 0\]
\[ \Rightarrow - x - 1 < 0 \left[ \because e^{- x} > 0, \forall x \in R \right]\]
\[ \Rightarrow - x < 1\]
\[ \Rightarrow x > - 1\]
\[ \Rightarrow x \in \left( - 1, \infty \right)\]
\[\text { So, f(x) is decreasing on }\left( - 1, \infty \right).\]
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