Exercises 40–49 concern the implicit function theorems and the inverse function theorem (Theorems 6.5, 6.6, and 6.7).
Let F(x, y) = x2 − y3, and consider the curve C defined by the equation F(x, y) = 0.
(a) Show that (0, 0) lies on C and that Fy(0, 0) = 0.
(b) Can we describe C as the graph of a function y = f (x)? Graph C.
(c) Comment on the results of parts (a) and (b) in light of the implicit function theorem (Theorem 6.5).
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