Explanation for behaviour of graph of $y=x^2e^{-x^2}$ (Maxwell-Boltzmann distribution) Consider the function
$$y=x^2e^{-x^2}$$
The graph initially behaves as a parabola then in later part exponential part of it dominates; i.e., the graph looks exponential after maximum of the curve.
Actually this graph is related to Maxwell Boltzmann distribution graph. Please help me so that I can easily remember the property of this graph.

$$f(x)=x^2e^{-x^2}\implies f'(x)=2x(1-x^2)e^{-x^2}, f''(x)=2x^2(x^4-5x^2+1)e^{-x^2}.$$
1- Check that $f(x)$ is even and $f(0)=0, f(\pm \infty)=0,. f'(x)=0 \implies x=0, \pm 1, f''(\pm 1)<0, f'(x)$ does not change sign around $x=0$. So min at $x=0$ max at $x=\pm 1, f_{max}=e^{-1}, f_{min}=0.$ All these information helps plotting this function as ny @Ross Millikan in his answer here.
