Study of x times e to the minus x

Text

Study the function $y=x\,e^{-x}$: domain, sign, limits at the ends, relative extrema and inflection.

Solution

Domain $\mathbb{R}$. Sign: $y>0$ for $x>0$, $y<0$ for $x<0$, zero at $x=0$. Limits: $\displaystyle\lim_{x\to+\infty}xe^{-x}=0^{+}$ (asymptote $y=0$), $\displaystyle\lim_{x\to-\infty}xe^{-x}=-\infty$. First derivative: $y'=e^{-x}(1-x)=0\Rightarrow x=1$: max $\big(1,\tfrac1e\big)\approx(1,0.368)$. Second derivative: $y''=e^{-x}(x-2)=0\Rightarrow x=2$: inflection at $\big(2,\tfrac{2}{e^{2}}\big)\approx(2,0.271)$. $\boxed{\text{max }(1,\tfrac1e);\ \text{inflection }(2,\tfrac{2}{e^{2}});\ \text{asymptote }y=0}$