Let $f:\mathbb{R^2}\rightarrow \mathbb{R}$, $f(x,y)=x^2+xy$. Determine all extreme points and characterize them as maxima or minima.
I computed the relative partial derivatives of the function as $f_{x}=2x+y$, $f_y=x$ $f_{xx}=2$, $f_{yy}=0$, $f_{xy}=1$. I know that If $f_{xx}f_{yy}-f_{xy}^2 <0$ at the points $(a,b)$ then the point is considered a saddle point, and that $f_x(0,0)=0, f_y(0,0)=0. $ So considering the second partial derivatives are both constants then the point $(0,0)$ is the only interesting point and it's the saddle point which means the function has no maxima or minima. Is this reasoning correct or is there something I'm missing ?
Yes, the reasoning is correct. But, in order to prove that $f$ has no maximum and no minimum, you could simply say that it has no maximum because $\lim_{x\to\infty}f(x,0)=\infty$ and that it has no minimum because $\lim_{x\to\infty}f(-1,x)=-\infty$.