Algebra Smallest Possible Value

Question

What is the smallest possible value of

$x^2+y^2-x-y-xy$?

Is this even possible to solve? Please help.

Answer 1

$$2(x^2+y^2-x-y-xy)=(x-y)^2+(x-1)^2+(y-1)^2-1-1\ge-2$$

Alternatively, let $$x^2+y^2-x-y-xy=k\iff x^2-x(1+y)+y^2-y-k=0$$

As $x$ is real, the discriminant must be $\ge0$

$$\implies(1+y)^2\ge4(y^2-y-k)$$

$$\iff4k\ge3y^2-6y-1=3(y-1)^2-3-1\ge-3-1$$

Answer 2

Conside the function$$f=x^2+y^2-x-y-xy$$ and its partial derivatives $$\frac{\partial f}{\partial x}=2x-1-y$$ $$\frac{\partial f}{\partial y}=2y-1-x$$ Since you look for an extremum, set them equal to $0$ to get $x=y=1$. At this point $f=-1$.

Could it be a maximum ?