Answer
Figure 11 (a):
$f$ has a local maximum at $A$ subject to the constraint.
Figure 11 (b):
$f$ has neither a local minimum nor a local maximum at $B$.
Work Step by Step
Consider Figure 11 (a):
Suppose that we move in the direction of northeast, $f$ is increasing (either we approach $A$ from the left or from the right) until we arrive at the point $A$, where $\nabla {f_A}$ is orthogonal to the constraint curve $g\left( {x,y} \right) = 0$. Once at $A$, we cannot increase $f$ further without leaving the constraint curve. Thus, $f$ has a local maximum at $A$ subject to the constraint.
Consider Figure 11 (b):
In this case, if we approach $B$ from the left, $f$ is increasing until we arrive at the point B. However, if we approach $B$ from the right, $f$ is decreasing until we arrive $B$. Therefore, we conclude that $f$ has neither a local minimum nor a local maximum at $B$.
