= Solution
Assume an identifiable smooth parametric model, an interior true parameter, a consistent interior <maximum-likelihood estimator>, interchange of differentiation and integration, a uniform local law of large numbers for the Hessian, and nonsingular finite <Fisher information>. For one observation define the <score function> $s_\theta(X)=\nabla_\theta\log p_\theta(X)$ and
$$
\boxed{J(\theta)=\mathbb E_\theta[s_\theta s_\theta^{\mathsf T}]
=-\mathbb E_\theta[\nabla_\theta^2\log p_\theta(X)]}.
$$
The <score function> has expectation zero by differentiating the normalization of the density. Its sample sum satisfies $n^{-1/2}S_n(\theta)\xrightarrow d N(0,J)$ by the vector <central limit theorem>. The <score function> equation and Taylor's formula give
$$
0=S_n(\theta)+\left[\int_0^1H_n(\theta+t(\widehat\theta_n-\theta))dt\right](\widehat\theta_n-\theta).
$$
The bracket divided by $n$ converges in probability to $-J$ by consistency and the assumed uniform law. Solving the equation and applying <Slutsky theorem> yields
$$
\boxed{\sqrt n(\widehat\theta_n-\theta)\xrightarrow d N(0,J(\theta)^{-1})}.
$$
The integral Hessian avoids incorrectly assuming a single intermediate point for a vector-valued Taylor formula.
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