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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.

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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 $x=1$.

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$y=e^{-x^2}=xe^{-x^2}$ implies that $x=1$ simplify by $e^{-x^2}$ and $y=e^{-1}$.

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