Here comes my question, while solving the optimization problem \begin{equation} \begin{aligned} \min\quad &x^TQx+q^Tx\\ & x=(x_1,x_2)^T\in\{(x_1,x_2)^T|x_1,x_2\geq0,x_1+x_2\leq x_0\} \end{aligned} \end{equation} $Q\in S^{2\times2}$ and the eigenvalues of which, one is positive and one is negative. The feasible region is triangle in the first quadrant.
Here's my ideas. First, the feasible is compact and the objective is continuous, so there must be optimal. $Q$ is symmetric which means we can always find orthogonal rotation to make the objective function separate about the $x_1,x_2$, and also the feasible region rotates with the objective. And Assume that the optimal is in the internal and find contradictions. But I'm doubtful about this method, so how to prove the optimal solution must lies on the boundary or not technically?