Finding the global max and min of a function on the region of a circle

Question

Let $D = [(x,y) \in \mathbb{R}: x^2+y^2\leq 4]$ Which is a circle of radius 2 centred at the origin, together with its interior. This is a compact region so any continous function on $D$ will attain a global max/min. Find the global max and global min of $f(x,y,) = x^2+xy+y^2$ on $D$.

Answer 1

HINT: solve $$\frac{\partial f(x,y)}{\partial x}=2x+y=0$$ and $$\frac{\partial f(x,y)}{\partial y}=2y+x=0$$ and then on the boundary $$x^2+y^2=4$$