Question

The function f(x) = x² e^−x is monotonic increasing when
(a) x ∈ R − [0, 2]
(b) 0 < x < 2
(c) 2 < x < ∞
(d) x < 0

Answer 1

(b) 0 < x < 2

$$\begin{gathered} f\left(x\right)=x^2{e}^{-x} \\ f\text{'}\left(x\right)=2x{e}^{-x}-x^2{e}^{-x} \\ =e^{-x}x\left(2-x\right) \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be monotonic increasing, we must have} \\ f\text{'}\left(x\right)>0 \\ \Rightarrow e^{-x}x\left(2-x\right)>0\left[\because e^{-x}>0\right] \\ \Rightarrow x\left(2-x\right)>0 \\ \Rightarrow x\left(x-2\right)<0 \\ \Rightarrow 0<x<2 \end{gathered}$$