= Solution
The per-observation <Fisher information> is $I(\theta)=\mathbb E_\theta[(\partial_\theta\log f(X_1,\theta))^2]$. Since
$$
\partial_\theta^2\log f=\frac{f_{\theta\theta}}f-\left(\frac{f_\theta}f\right)^2,
$$
and the <integral> of $f_{\theta\theta}$ is the second <derivative> of $1$, the regularity assumptions give
$$
\boxed{I(\theta)=-\mathbb E_\theta[\partial_\theta^2\log f(X_1,\theta)].}
$$
For independent observations the total <Fisher information> is $nI(\theta)$, because centered individual <score functions> have zero cross terms.
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