Study of x·e to the −x

Problem

Study $f(x) = x\,e^{-x}$ (domain $\mathbb{R}$, all steps, with graph).

Solution

Domain: $\mathbb{R}$. Symmetry: none. Intersections: $f(0)=0$, the only zero at $x=0$; sign agreeing with $x$ (since $e^{-x}>0$).

Asymptotes: $\lim_{x\to+\infty} x e^{-x} = 0^+$ (horizontal asymptote $y=0$ on the right); $\lim_{x\to-\infty} x e^{-x} = -\infty$.

First derivative: $f'(x)=e^{-x}-x e^{-x}=e^{-x}(1-x)$. It vanishes at $x=1$; $f'>0$ for $x<1$, $f'<0$ for $x>1$: maximum at $\bigl(1,\tfrac1e\bigr)$, with $\tfrac1e\approx 0{,}368$.

Second derivative: $f''(x)=e^{-x}(x-2)$, vanishes at $x=2$: inflection point at $\bigl(2,\tfrac{2}{e^2}\bigr)$.

Maximum at $(1,1/e)$, asymptote $y=0$ on the right, divergence to $-\infty$ as $x\to-\infty$.

Links

Topics: Derivatives
Concepts: Asymptote · Maximum minimum · Rules of differentiation
Skills: Study a function · Sketch a graph
Exercise type: Study of a function