Covariance of sample mean and slope

Consider the simple linear regression model $y_i = \beta_0 + \beta_1 x_i + \epsilon_i$, $i = 1,2,…,n$, where $\beta_0$ and $\beta_1$ are unknown parameters, $\epsilon_i$'s are uncorrelated random errors with mean 0 and finite variance $\sigma^2 > 0$. Let $\bar{y} = \frac1n \sum_{i=1}^n y_i$ and $\hat\beta_1$ be the least squares estimator of $\beta_1$. Then which one of the following statements is true?