To minimize $x^TAx$ where $A$ is not necessarily positive semi-definite.

Suppose that there is some $v$ such that $v^TAv < 0$, i.e. $A$ is not positive semidefinite. Then for every $\lambda > 0$, we have $$(\lambda v)^TA(\lambda v) = \lambda^2 (v^T Av) \ \overset{\lambda \to \infty}{\longrightarrow}\ -\infty$$ It follows that if $A$ is not positive semidefinite the problem is unbounded from below. This is in particular true for indefinite matrices.