EXERCISE 4.28. Verify that the curvature at the origin of the monkey saddle, S = {(u, v, u³ - 3v2u) | u, v E R2}, equals zero; see Fig. 4.11. In Proposition 4.14, can any conclusion be drawn when K (p) = 0? FIGURE 4.11. The monkey monkey saddle {(u, v, u³ - 3v²u) | u, v Є R²} PROPOSITION 4.14. Let S be a (not necessarily oriented) regular surface, and let pЄ S. If K(p) > 0, then a sufficiently small neighborhood of p in S lies entirely on one side of the plane p+TpS (except for the point p itself, which lies in this plane). If K(p) < 0, then every neighborhood of p in S intersects both sides of p+TpS. PROOF. As in the proof of Lemma 4.5 on page 200, after applying a rigid motion, we can assume without loss of generality that TpS = span{e1, e2} and that a neighborhood of p in S equals the graph of a smooth function f. According to Example 4.4 on page 198, K(p) = fxxfyy-fy. The result now follows from the second derivative test from multivariable calculus, which classifies the critical point as a (strict) local extremum if fxxfyy - fay > 0 and as a saddle point if fær fyy - fy<0.