Bernoulli information identities hold, then the bound can vanish

Let $X_1,\ldots,X_n$ be independent $\operatorname{Bernoulli}(p)$ observations with $0<p<1$ unknown, and write $S=\sum_{i=1}^n X_i$. Starting from the joint log-likelihood, derive the score $U_n(p)$. Prove the three regular identities $$ \mathbb E_p[U_n(p)]=0, \qquad I_n(p)=\operatorname{Var}_p(U_n(p)) =-\mathbb E_p[\partial_p U_n(p)], $$ and compute $I_n(p)$ explicitly. Deduce the Cramér–Rao lower bound for every unbiased estimator of $p$, and show that $\bar X$ attains it. Now let $\tau(p)=p(1-p)$. Compute $\tau'(p)$ and the Cramér–Rao expression for unbiased estimators of $\tau(p)$. Evaluate that expression at $p=1/2$ and explain why a vanishing bound does not produce a zero-variance estimator of $1/4$. Contrast this regular, fixed-support calculation with a model in which the support moves with the parameter: the identities above need not hold there, but do not re-derive a shifted-exponential example. Do not construct a most-powerful test of $H_0:p=p_0$.

The log-likelihood is $S\log p+(n-S)\log(1-p)$. Differentiate once and twice.