= Solution
This is a symmetric <Beta distribution> with parameters $(2,2)$, so its mean and <median> are both $1/2$. At the <median> its <probability density function> is $f(1/2)=3/2$, giving <asymptotic variance> $1/9$ for the <sample median>. Direct integration gives
$$
\mathbb EX=\frac12,\qquad \mathbb EX^2=6\int_0^1x^3(1-x)\,dx=\frac3{10},\qquad \operatorname{Var}(X)=\frac1{20}.
$$
Consequently $\sqrt n(\bar X_n-1/2)\xrightarrow{d}N(0,1/20)$. \b[The <sample median> has $20/9$ times the <asymptotic variance> of the <sample mean>], with <asymptotic relative efficiency>
$$
\boxed{\operatorname{ARE}(\text{median},\text{mean})=\frac9{20}.}
$$
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