Answer
Absolute minimum: $f(1,0)=-1$
Absolute maximum: $f(0,2)=f(0,-2)=4$
Work Step by Step
To calculate the critical points we have to first put $f_x(x,y), f_y(x,y)$ equal to $0$.
Thus, $f_x=2x-2,f_y=2y$
This yields, $x=1,y=0$ and $f(0,1)=-1$
From the vertices of the triangle, after simplifications, we have:
$x=x ,y=2-x,x-2$
Now, $f(x,2-x)=x^2+(2-x)^2-2x=2x^2-6x+4$ and
$f(x,x-2)=x^2+(x-2)^2-2x=2x^2-6x+4$
Also,
$f'(x,2-x)=4x-6$ and $f'(x,x-2)=4x+6$
This gives $x=\dfrac{3}{2},\dfrac{3}{2}$,$y=\dfrac{1}{2},-\dfrac{1}{2}$
Hence,
Absolute minimum: $f(1,0)=-1$
Absolute maximum: $f(0,2)=f(0,-2)=4$
