I am studying for a fast approaching Calc 3 midterm exam and ran into this problem in the textbook.
Show that $f(x,y) = x^2+4y^2-4xy+2$ has an infinite number of critical points and that $D = 0$ at each one. Then show that $f$ has a local (and absolute) minimum at each critical point.
So I am able to solve the two partial derivatives to get $f_x=2x-4y$ and $f_y=8y-4x$ and that I can solve both equations for $f_x=f_y=0$ which is true for all points. I also can see that this is is true for all values: $$D = (f_{xx})(f_{yy})-(f_{xy})^2 = 0$$
However, I dont know how to go about determining that all of the critical points are local and absolute minimums. I believe that I was only taught the 2nd derivative test where you compare the value of $D$ to $f_{xx}$.
Any ideas? Thanks!