Consider the surface .# in R' defined by the equation x> P +2ze* = 0. 2 (a) Show that, in a neighbourhood of the point (1,1,0), the surface .# can be written as z = g(x,y) for a C! function g. Compute gi(l,l) and %(1, 1). (b) Now consider the surface .% near the (0,0, 0). (b1) Show that, in a neighbourhood of (0,0, 0), we can write .# as z = h(x, y) for a C! function h oh and compute Ix (0,0) and g(, 0). (b2) Directly from the equation defining .#, write x = p(y,z) and y = g(x, z) for some functions p and gq. Explain why such functions p and g cannot be obtained from the Implicit Function Theorem