Study of x squared times e to the minus x

Text

Study the function $y=x^{2}e^{-x}$: domain, sign, limits, relative extrema.

Solution

Domain $\mathbb{R}$; $y\ge0$, zero at $x=0$. Limits: $\displaystyle\lim_{x\to+\infty}x^{2}e^{-x}=0^{+}$ (asymptote $y=0$), $\displaystyle\lim_{x\to-\infty}x^{2}e^{-x}=+\infty$. First derivative: $y'=x e^{-x}(2-x)=0\Rightarrow x=0,\ x=2$. $x=0$ minimum (value $0$); $x=2$ maximum $\big(2,\tfrac{4}{e^{2}}\big)\approx(2,0.541)$. $\boxed{\text{min }(0,0);\ \text{max }\big(2,\tfrac{4}{e^{2}}\big)\approx(2,0.541)}$