Attached the level curves map for the objective function as well as the stationary points obtained applying the Lagrange multipliers technique to solve this problem. In red we can observe the gradient direction for the restrictions at the stationary points as well as the gradient direction for the objective function at the same points (in black).
Points $B, C, E, D$ are the local extrema. Inside unlabeled, we can observe a saddle point. Point $A$ is not a feasible local extrema because the objective function's gradient (black) is not a positive combination of the corresponding restriction's gradient.
The plot shows the maximum determination case.
& x & y & f(x,y) \\
D & 0 & -8 & 36 \\
& 0 & -2 & 0 \\
A & 0 & 0 & 4 \\
E & 2 & -2 & 4 \\
C & 8 & 0 & -28 \\
B & 2 & 0 & 8 \\
Depending on maximum or minimum the objective gradient vector should be considered with opposite sign. According to that, points $C,E$ as shown are not feasible but considering a minimization $-\nabla f$ they could be feasible.