show that the function

has a unique critical point. what kind is it?

I tried using the second derivative test. So from first derivatives I got the critical points (0,0) (1,0) and (1,2)

How? is the only solution to this system, as can be checked...or better: check that are not zeroes of this sytem but [tex]

I then tried using the determinant where

but it was always inconclusive. For example D=0 for the point (0,0) and

for (1,0) and I don't know how to interpret that.

And you don't need to: critical points are points where the first order partial derivatives vanish, and that's what's been done above
I haven't learned La

grange multipliers yet.