Study of y = x²·e^(−x)

Text

Study $y=x^{2}e^{-x}$: sign, behaviour at $\pm\infty$, maxima and minima, inflections.

Solution

Sign. $y=x^{2}e^{-x}\ge0$ for all $x$, with a double zero at $x=0$. Limits. $\displaystyle\lim_{x\to+\infty}x^{2}e^{-x}=0^{+}$ (horizontal asymptote $y=0$ on the right); $\displaystyle\lim_{x\to-\infty}x^{2}e^{-x}=+\infty$. Extrema. $y'=x e^{-x}(2-x)$, vanishing at $x=0$ and $x=2$. At $x=0$ minimum $y=0$; at $x=2$ maximum $y=4e^{-2}\approx0.5413$. Inflections. $y''=e^{-x}\left(x^{2}-4x+2\right)$ vanishes at $x=2\pm\sqrt2$: two inflections. $\boxed{\text{min }(0,0);\quad \text{max }\left(2,\,4e^{-2}\right)\approx(2,\,0.5413)}$