Optimal points of $f(x,y)=x^2 + y^2 + \beta xy + x + 2y$
If the function is convex, then all local minimum are global minimum.
If $4-\beta^2 &gt;0$, then the function is convex and hence the local minimum is indeed the global minimum.
If $\beta = \pm 2$, then we have $f(x,y)=(x\pm y)^2+x+2y$, $f(x, \mp x)=x\mp2x$ of which we can make it arbitrary large or small.
If $4-\beta^2 &lt; 0$, then it is indefinite, the stationary point is a saddle point.
It can't be negative definite, it doesn't have global maximum.