# Asymptotic consistency and normality

I need help getting the following problem

Let $$X_1,..,X_n$$ be independent $$N(\mu,1)$$-distributed random variables. Define $$\hat{\theta_n}$$ as the point of minimum of $$\sum_{i=1}^n(X_i-\theta)^4$$.

(1) Show that $$\hat{\theta}_n \rightarrow \theta_0$$ for some $$\theta_0$$.
(2) Show that $$\sqrt{n}(\theta_n - \theta_0)$$ converges to normal distribution.

For (1) I have no idea what $$\theta$$ is and how to start. For (2) I know I have to use asymptotic normality where

$$\sqrt{n}(\hat{\theta}_n - \mu) \rightarrow N \left(0, \frac{P\psi_{\theta_0}}{(P\psi_{\theta_0})^2}\right)$$ where $$P()$$ stands for an expectation of sorts, but I'm not sure how that one is evaluated. Any help or hints will be appreciated