Test the uniform convergence of $x^n-x^{2n}$ in $[0,1]$

Question

$$ f_n(x)=x^n-x^{2n} \\ f_n:[0,1]\rightarrow \mathbb R $$

I know that the function $x^n$ is not converging uniformally because the limiting function is not continuous (when x=1 there's a "step" in the graph). Here There exists no such step. I'm thinking about using Weierstrass' M-test but not sure which $M_n$ to pick.

Any hints?

Answer 1

Note that $f$ is differentiable and $f'(x)=nx^{n-1}-2nx^{2n-1}$ which means that $f$ has extrema at

$$n-2nx^{n}=0$$

or $(1/2)^{1/n}$. As a commenter noted, $f(x)\ge 0$ on that interval so you have a maximum. If you plug in the value of $x$, does the bound go to zero as $n$ grows?