Question

If f Subscript x​(a,b)equals f Subscript y​(a,b)equals​0, 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.

Answer 1

D.

Explanation:

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.