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Consider a Lipschitz continuous function $f$ on $[0,1]$ with constant $L_f$, and its Bernstein polynomial $B_{n,f}$ of degree $n$. It can be shown that

$$\sup_{x\in[0,1]}|B_{n,f}(x) - f(x)| \leq \frac{L}{2\sqrt{n}}.$$

However, I read from somewhere that if $f$ has Lipschitz continuous derivative $g$ with constant $L_g$, then the bound is improved to be $O(\frac{1}{n})$. How is this bound derived explicitly?

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