# Find the points of intersection between two curves $y=e^{-x^{2}}, y = xe^{-x^2}$

Find the points of intersection between two curves: $$y=e^{-x^{2}}, y = xe^{-x^2}$$

Idea:

The basic observation suggests one of point $$x=1$$, $$y = e^{-1}$$

But I am not sure how proof this only solution.

That is the only solution, since $$e^{-x^2}=xe^{-x^2}\implies e^{-x^2}(x-1)=0,$$ and since $$e^{-x^2}\neq 0$$ for all $$x,$$ it follows that $$(x-1)=0$$, i.e., that $$x=1$$.
$$y=e^{-x^2}=xe^{-x^2}$$ implies that $$x=1$$ simplify by $$e^{-x^2}$$ and $$y=e^{-1}$$.