= Solution
The <score function> is
$$
S_n(\theta)=\frac{d}{d\theta}\log\prod_{i=1}^nf(X_i,\theta)=\sum_{i=1}^n\frac{\partial_\theta f(X_i,\theta)}{f(X_i,\theta)}.
$$
With common parameter-independent support and justified differentiation under the <integral>, $\mathbb E_\theta[\partial_\theta\log f(X_1,\theta)]=\int\partial_\theta f(x,\theta)dx=\partial_\theta1=0$. Linearity then gives $\boxed{\mathbb E_{\theta_0}S_n(\theta_0)=0}$. These regularity conditions exclude, for example, moving-support uniform models.
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