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.