I'm having some trouble understanding the following thing my teacher wrote:
Let $f(x,y)=\cos(x + 2y) + \cos(2x + y)$, then: $$\text{hess}(f)|_{(0,0)}=\left(\begin{matrix} -5 & -4 \\ -4 & -5\end{matrix} \right)$$
The eigenvalues are $-1$ and $-9$ and an orthonormal eigenbasis is: $(\frac{1}{\sqrt 2}(1,1),\frac{1}{\sqrt 2}(1,-1)) $.
This means that locally, near the point $(0,0)$, the function $f$ resembles the function $1 -\frac 9 2 x^2 - \frac 1 2 y^2$.
So we can conclude that the function $f$ has a local maximum at the point $0,0$.
I don't get how my teacher concluded that the function resembles the function $1 -\frac 9 2 x^2 - \frac 1 2 y^2$ near the origin. How can I conclude that?