function f = green level curves, constraint g = pink curve
I'm supposed to identify what point A and B are in the function f. The options are (a) local max (b) local min (c) neither.
For A, I think it is a local minimum. For B, I think it is neither.
Can someone tell me if I'm wrong or right? If wrong, why? Thank you.
$A$ is a local maximum. Indeed, the gradient is orthogonal to the tangent of the pink curve, hence its restriction to that curve is $0$. Furthermore, it reaches the level curve at only one point without crossing it. (The pink curve is always in the zone such that $f(x,y) \le 2$ and $f(A)=2$.)
For the second case, we have almost the same situation excepted that the pink curve crosses the level curve. Hence $B$ is neither a local max or a local min. At the left of $B$, we have $f(x,y)<3$ and at the right, $f(x,y)>3$.