So I have the following question: Suppose that the Hessian matrix of a certain quadratic form $f(x, y)$ is $ H = \begin{bmatrix} 4 & 8 \\ 8 & 4 \end{bmatrix}$
Find the critical point and the classification of the critical point of f, assume $λ1<λ2$.
So I solved for the eigenvalues of the Hessian Matrix and got 12 and -4. I know he critical point is a relative minimum, however how do I solve for the critical point?
I would look into theorems 3.5 & 3.6 in the link I'm including (Link:http://emp.byui.edu/brownd/mathematics/calculus-rn-rm/hessian-etc.pdf).
These theorems give information on how to use the eigenvalues to determine the concavity of the surface and classify a given critical point. Notice I said "classify" a critical point because the eigenvalues give us no information on the location of the critical points, only whether those critical points would be minima, maxima, or saddle points. You are correct in this case that there should only be one critical point because a positive Hessian implies that $f(x,y)$ is a quadratic polynomial; however, it still doesn't allow us to find the critical point, as there are many possible $f(x,y)$ that would satisfy this Hessian. For example $$f(x,y)= 2(x^2+4xy+y^2)+ax+cy+d$$ would have this Hessian for any constants $a,c,$ and $d$.