For the function

f$($x$,$ y$)=2$x$^2-2$xy $+$ y$^2-$ y $+3$

a$)$ Find any ordinary critical points and determine whether they are maxima, minima or saddle points.

b$)$ On the closed triangle R with vertices $(0,0),(2,0)$ and $(0,2)$ find any possible local maxima or minima on the boundary of R$.$

c$)$ Using the results from part a$)$ and b$),$ find the global $($absolute$)$ maximum and minimum of f$($x$,$ y$)$ and give the values of f$($x$,$ y$)$ at these points.

The second derivative test for f$($x$,$ y$),$ where d$1=$ fxx and d$2=$ det Hf

d$2>0$ and d$1>0$: Local Minimum

d$2>0$ and d$1<0$: Local Maximum

d$2<0$ : Saddle Point

d$2=0$ : Can't determine type by this method