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