Answer
1. Critical point $A$, a local maximum.
2. Critical point $B$, a saddle point.
3. Critical point $C$, a local minimum.
4. Critical point $D$, a saddle point.
Work Step by Step
1. Critical point $A$
No matter which direction we walk, it will take us downhill, so it is a local maximum.
2. Critical point $B$
If we walk in the directions $ + {\bf{i}}$ or $ - {\bf{i}}$, it will take us uphill. If we walk in the directions $ + {\bf{j}}$ or $ - {\bf{j}}$, it will take us downhill. Therefore, we conclude that it is a saddle point.
3. Critical point $C$
No matter which direction we walk, it will take us uphill, so it is a local minimum.
4. Critical point $D$
If we walk in the directions $ + {\bf{i}}$ or $ - {\bf{i}}$, it will take us downhill. If we walk in the directions $ + {\bf{j}}$ or $ - {\bf{j}}$, it will take us uphill. Therefore, we conclude that it is a saddle point.
