Question

Consider the quadratic form $$q\left(x_{1}, x_{2}\right)=a x_{1}^{2}+b x_{1} x_{2}+c x_{2}^{2}$$ We define $$q_{11}=\frac{\partial^{2} q}{\partial x_{1}^{2}}, \quad q_{12}=q_{21}=\frac{\partial^{2} q}{\partial x_{1} \partial x_{2}}, \quad q_{22}=\frac{\partial^{2} q}{\partial x_{2}^{2}}$$ The discriminant $D$ of $q$ is defined as $$D=\operatorname{det}\left[\begin{array}{ll}q_{11} & q_{12} \\q_{21} & q_{22}\end{array}\right]=q_{11} q_{22}-\left(q_{12}\right)^{2}$$ The second derivative test tells us that if $D$ and $q_{11}$ are both positive, then $q\left(x_{1}, x_{2}\right)$ has a minimum at (0,0) Justify this fact, using the theory developed in this section.

          Consider the quadratic form $$q\left(x_{1}, x_{2}\right)=a x_{1}^{2}+b x_{1} x_{2}+c x_{2}^{2}$$ We define $$q_{11}=\frac{\partial^{2} q}{\partial x_{1}^{2}}, \quad q_{12}=q_{21}=\frac{\partial^{2} q}{\partial x_{1} \partial x_{2}}, \quad q_{22}=\frac{\partial^{2} q}{\partial x_{2}^{2}}$$ The discriminant $D$ of $q$ is defined as $$D=\operatorname{det}\left[\begin{array}{ll}q_{11} & q_{12} \\q_{21} & q_{22}\end{array}\right]=q_{11} q_{22}-\left(q_{12}\right)^{2}$$ The second derivative test tells us that if $D$ and $q_{11}$ are both positive, then $q\left(x_{1}, x_{2}\right)$ has a minimum at (0,0) Justify this fact, using the theory developed in this section.

Added by Brittany B.

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