The Hessian matrix
The Hessian matrix of \( C(\beta) \) is given by
$$
\hat{H} \equiv \begin{bmatrix}
\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\
\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\
\end{bmatrix} = 2X^T X.
$$
This result implies that \( C(\beta) \) is a convex function since the matrix \( X^T X \) always is positive semi-definite.