I'm still waiting for your partial derivatives and why you think that bore no fruit. Did you investigate those boundaries, yet?
Here's how to classify the critical points. I think you got all of them...how did you get (0,1/2)?
where is a critical point.
Here are the rules:
1. If and , then is a local minimum
2. If and , then is a local maximum
3. If , then is a saddle point (neither a max or min)
those are all the points you got from the boundary?