If f Subscript x(a,b)equals f Subscript y(a,b)equals0, does it follow that f has a local maximum or local minimum at (a,b)?
Explain. Choose the correct answer below. A. Yes. The point (a,b) is a critical point and must be a local maximum or local minimum. B. Yes. The tangent plane to f at (a,b) is horizontal. This indicates the presence of a local maximum or a local minimum at (a,b). C. No. One (or both) of f Subscript x and f Subscript y must also not exist at (a,b) to be sure that f has a local maximum or local minimum at (a,b). D. No. It follows that (a,b) is a critical point of f, and (a,b) is a candidate for a local maximum or local minimum.
The point must be a critical point but it could be a saddle point. If the point is a saddle point it would not be neither a maximum nor a minimum. So it must be critical but it does not follow directly that it has a local maximum or local minimum.
Therefore D. (a,b) would be a candidate, but is not necessarily a maximum or minimum. It could be a saddle.
