Chapter 12.8, Problem 61E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume that f is differentiable at the points in question.

a.    The fact that f_x(2, 2) = f_y(2, 2) = 0 implies that f has a local maximum, local minimum, or saddle point at (2, 2).

b.    The function f could have a local maximum at (a, b) where $f_y\left(a,b\right)\ne 0.$

c.    The function f could have both an absolute maximum and an absolute minimum at two different points that are not critical points.

d.    The tangent plane is horizontal at a point on a smooth surface corresponding to a critical point.