Suppose that F(u,v) is of class C1 and is such that F(−2, 1) = 0 and Fu (−2, 1) = 7, Fv(−2, 1) = 5. Let G(x, y, z) = F(x3 − 2y2 + z5, xy − x2z + 3).
(a) Check that G(−1, 1, 1) = 0.
(b) Show that we can solve the equation G(x, y, z) = 0 for z in terms of x and y (i.e., as z = g(x, y), for (x, y) near (−1, 1) so that g(−1, 1) = 1).
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