(Solved): 6. Determine if the following quadratic forms are positive definite, positive semidefinite, negati ...

6. Determine if the following quadratic forms are positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite. (a) \( f(\vec{x})=\vec{x}^{T}\left[\begin{array}{cc}-4 & 2 \\ 0 & -2\end{array}\right] \vec{x} \) (b) \( g(\vec{x})=\vec{x}^{T}\left[\begin{array}{ccc}2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right] \vec{x} \) (c) \( h\left(x_{1}, x_{2}, x_{3}\right)=x_{1}^{2}+2 x_{2}^{2}-x_{1} x_{3} \)