Question: (a) The graph of f (x, y) = (y - x2)(y - 2x2) intersects the xy-plane z = 0 in two parabolas. In the xy-plane, draw the domains where f is negative, and where f is positive. Show that (0, 0) is the only stationary point, and that it is a saddle point.
(b) Suppose (h, k) ≠ (0, 0) is any direction vector. Let g(t) = f (th, tk) and show that g has a local minimum at t = 0, whatever the direction (h, k) may be. (Thus, although (0, 0) is a saddle point, the function has a local minimum at the origin in each direction through the origin.)