real analysis - Is the sequence of functions $f_n(x) = nxe^{-nx^2}$ uniformly bounded on the interval $[0,1]$?

Is the sequence of functions $f_n(x) = nxe^{-nx^2}$ uniformly bounded on the interval $[0,1]$?

I have been trying to figure out if it is uniformly bounded on $[0,1]$ but am having trouble verfiying it. I know that by definition if for ALL $x \in [0,1]$ and ALL $n$ I can show that $f_n(x)$ is bounded above by some constant, I am good to go. But, it seems that the $x$ portion of the function is bounded ok but the $n$ part may not be. Does anyone have a good way of showing if it is uniformly bounded? thank you!

Answer

Let's find the maximum of $f_n(x)$: Notice that $f_n'(x)=ne^{-nx^2}-2n^2x^2e^{-nx^2}$, so $f_n'(x)=0\iff n=2n^2x^2\iff x=\sqrt{1/2n}$, and this is in $[0,1]$.

Since $f_n(\sqrt{1/2n})=n\sqrt{1/2n}e^{-n(1/2n)}=\sqrt{n/2}e^{-1/2}\to\infty$, then $(f_n)$ is not uniformly bounded.