Solution: The correct answer is ts the forth one which is the contour plot of f(x,y) = x y. The one above is f(x,y) = x²+y² which has f_xy=0 at the center. The first one is just f(x,y)=g(x) for some function of x, like x², there is no y dependence so that f_xy=0 there too. In the third case, f_x is constant zero on the y axes as these are maxima.

Math21b | Math 21a | Fall 2008 | Department of Mathematics | Faculty of Art and Sciences | Harvard University