Question

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

Answer 1

$$\begin{gathered} f\left(x\right)=\left(x+2\right)e^{-x} \\ f\text{'}\left(x\right)=-e^{-x}\left(x+2\right)+e^{-x} \\ =-x{e}^{-x}-2e^{-x}+e^{-x} \\ =-x{e}^{-x}-e^{-x} \\ =e^{-x}\left(-x-1\right) \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be increasing, we must have} \\ f\text{'}\left(x\right)>0 \\ \Rightarrow e^{-x}\left(-x-1\right)>0 \\ \Rightarrow -x-1>0\left[\text{∵}{e}^{-x}>0,\forall x\in R\right] \\ \Rightarrow -x>1 \\ \Rightarrow x<-1 \\ \Rightarrow x\in \left(-\infty ,-1\right) \\ \text{So,}\mathit{\text{f}}\text{(}\mathit{\text{x}})\text{is increasing on}\left(-\infty ,-1\right). \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be decreasing, we must have} \\ f\text{'}\left(x\right)<0 \\ \Rightarrow e^{-x}\left(-x-1\right)<0 \\ \Rightarrow -x-1<0\left[\text{∵}{e}^{-x}>0,\forall x\in R\right] \\ \Rightarrow -x<1 \\ \Rightarrow x>-1 \\ \Rightarrow x\in \left(-1,\infty \right) \\ \text{So,}\mathit{\text{f}}\text{(}\mathit{\text{x}})\text{is decreasing on}\left(-1,\infty \right)\text{.} \end{gathered}$$