Our simple $2\times 2$ example

Last week we introduced the simple two-dimensional function

$$f(x_1,x_2)=x_1^2+x_1x_2+10x_2^2-5x_1-3x_2,$$

which is of the form (in terms of vectors and matrices)

$$f(\boldsymbol{x})=\frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x}-\boldsymbol{b}^T\boldsymbol{x},$$

where we have

$$\boldsymbol{x}=\begin{bmatrix} x_1 \\ x_2\end{bmatrix},$$ $$\boldsymbol{b}=\begin{bmatrix} 5 \\ 3\end{bmatrix},$$

and

$$\boldsymbol{A}=\begin{bmatrix} 2 & 1\\ 1& 20\end{bmatrix}.$$

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