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.
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$$f(x)=x^2e^{-x^2}\implies f'(x)=2x(1-x^2)e^{-x^2}, f''(x)=2x^2(x^4-5x^2+1)e^{-x^2}.$$ 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.
You actually gave the mathematical explanation. The graph is below. Over the range $[-1,1]$ the exponential doesn't change that much-it is $1$ at the center and $\frac 1e \approx 0.3679$ at the ends. That is less than a factor $3$. The parabola is $0$ at the middle and $1$ at the ends, an infinite ratio. It dominates the product over this interval. As you get outside that interval, the exponential dominates. From $1$ to $3$ the parabola rises by a factor $9$, but the exponential drops by a factor $2980$, so it dominates.