Find the intervals in which the function $f(x) = (x+2) e^{-x}$ is strictly increasing or decreasing.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Strictly increasing on $(−∞,−1]$ and strictly decreasing on $[−1,∞)$.
Given $f(x) = (x + 2) e^{-x}$, its domain = R.
Differentiating w.r.t. x, we get
$f'(x) = (x+2) e^{-x} (-1) + e^{-x} (1+0) = -(x + 1) e^{-x}$.
Now $f'(x) > 0$ iff $- (x + 1) e^{-x}>0$ but $e^{-x} > 0$ for all $x ∈ R$
$⇒ -(x + 1) > 0⇒x+1 <0⇒x<-1$
⇒ f(x) is strictly increasing in $(-∞, -1]$.
And $f'(x) < 0$ iff $- (x + 1) e^{-x} <0$ but $e^{-x} > 0$ for all $x ∈ R$
$⇒− (x + 1) <0⇒x+1>0⇒x>-1$
⇒ f(x) is strictly decreasing in $[-1, ∞)$.