Finding absolute points in a closed area

Question

$f(x,y)$ is $x^2+xy+y^2$. I need the max and min points for $f(x,y)$ in and on the rectangle $-2≤x≤2$ and $-1≤y≤1$.

What I did was;

$$f_x'=2x+y+y^2 = 0$$

$$f_y'=x^2+x+2y = 0$$

I couldn't implement the restriction of the rectangle.

Answer 1

First of all, your partial derivatives are wrong. You should treat the variable you are not differentiating as a constant, so you have:

$$\frac{\delta}{\delta x} = 2x + y$$ $$\frac{\delta}{\delta y} = x+2y$$

From this information, you can find the minimum value. Since the maximum value would be unbounded if there was no restriction, the maximum value must occur on the vertices of the region.