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Solution for Find the Intervals in Which F(X) = (X + 2) E−X is Increasing Or Decreasing ? - CBSE (Science) Class 12 - Mathematics

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Question

Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?

Solution

\[f\left( x \right) = \left( x + 2 \right) e^{- x} \]

\[f'\left( x \right) = - e^{- x} \left( x + 2 \right) + e^{- x} \]

\[ = - x e^{- x} - 2 e^{- x} + e^{- x} \]

\[ = - x e^{- x} - e^{- x} \]

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

\[\text { Forf(x) to be increasing, we must have }\]

\[f'\left( x \right) > 0\]

\[ \Rightarrow e^{- x} \left( - x - 1 \right) > 0\]

\[ \Rightarrow - x - 1 > 0 \left[ \because e^{- x} > 0, \forall x \in R \right]\]

\[ \Rightarrow - x > 1\]

\[ \Rightarrow x < - 1\]

\[ \Rightarrow x \in \left( - \infty , - 1 \right)\]

\[\text { So,f(x)is increasing on} \left( - \infty , - 1 \right) . \]

\[\text { Forf(x) to be decreasing, we must have }\]

\[f'\left( x \right) < 0\]

\[ \Rightarrow e^{- x} \left( - x - 1 \right) < 0\]

\[ \Rightarrow - x - 1 < 0 \left[ \because e^{- x} > 0, \forall x \in R \right]\]

\[ \Rightarrow - x < 1\]

\[ \Rightarrow x > - 1\]

\[ \Rightarrow x \in \left( - 1, \infty \right)\]

\[\text { So,f(x)is decreasing on }\left( - 1, \infty \right).\]

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Solution for question: Find the Intervals in Which F(X) = (X + 2) E−X is Increasing Or Decreasing ? concept: Increasing and Decreasing Functions. For the courses CBSE (Science), CBSE (Commerce), CBSE (Arts), PUC Karnataka Science
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