Largest covariance allowed by two variances

Random variables $X_1$ and $X_2$ have variances $24$ and $6$ respectively, while their means are $-3$ (for $X_1$) and $7$ (for $X_2$). What is the greatest value $\text{Cov}(X_1,X_2)$ can take?

$12$. By Cauchy–Schwarz (correlation lies in $[-1,1]$), $|\text{Cov}(X_1,X_2)|\le\sqrt{24\cdot6}=12$, with equality when $X_2$ is an increasing linear function of $X_1$. The means are irrelevant.