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

ConceptIncreasing and Decreasing Functions

#### 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 Find the Intervals in Which F(X) = (X + 2) E−X is Increasing Or Decreasing ? Concept: Increasing and Decreasing Functions.
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