Consider the equation $y_i=\beta_0+\beta_1x_i+\epsilon_i$ for $i=1, \dotsc, n$.
We have unbiased estimators $b_0$ and $b_1$ for $\beta_0$ and $\beta_1$ respectively, where $b_0=\bar{y}-b_1\bar{x}$ and $b_1= S_{xy} / S_{xx}$.
How does one show that $\operatorname{Cov}(b_0,b_1)=-\frac{\sigma\bar{x}}{S_{xx}}$
I tried using $\operatorname{Cov}(b_0,b_1)=E(b_0b_1)-E(b_0)E(b_1)$ to no avail as it just equals $0$ when I try and do that. Thanks!