Assuming sufficient regularity conditions on the likelihood $f(x \mid \theta)$ for a univariate parameter $\theta \in \Theta$, establish the Cramér-Rao lower bound for the variance of an unbiased estimator of $\theta$.

If $\hat{\theta}(X)$ is an unbiased estimator of $\theta$ whose variance attains the Cramér-Rao lower bound for every value of $\theta \in \Theta$, show that the likelihood function is an exponential family.