. (The implicit function theorem.) The implicit function theorem states that if F E C2 in some neighbourhood of (0,0), F(0,0) 0, and F, (0,0) 0, then there is a unique function y f(x), with f (0) 0, such that F(x,f(x)) 0 for x in a neighbourhood of 0. Prove this as follows: (a) Find a function y(x) which satisfies the equation F, (x,0 Fy (x, 0) for small x. Show that it will then follow that F(x,y(z)) (b) Write () as 0, y)+g(x,y), where f(0) g(z, 0) gy(x, 0) 0. (**) Solve (**) by successive approximations as follows: Let Yo = f(x) + g(x,0), Yn+1 f(x) + g(x,m), n 0 Show that, in a small neighbourhood of (0, 0), for example lxl -6, 11-5, in which the inequalities If(x)I 0/2, Ign(zw)1 1/2 hold, all the functions yn(x) satisfy ( (c) Hence, show that lyn(x)-ym(x)| m, and therefore Un(x) y(x) for lxl , where y(x) satisfies (**). Show the y(x) is unique, that is, if yi(x) is a second solution to (*), then y 1 (d) . (The implicit function theorem.) The implicit function theorem states that if F E C2 in some neighbourhood of (0,0), F(0,0) 0, and F, (0,0) 0, then there is a unique function y f(x), with f (0) 0, such that F(x,f(x)) 0 for x in a neighbourhood of 0. Prove this as follows: (a) Find a function y(x) which satisfies the equation F, (x,0 Fy (x, 0) for small x. Show that it will then follow that F(x,y(z)) (b) Write () as 0, y)+g(x,y), where f(0) g(z, 0) gy(x, 0) 0. (**) Solve (**) by successive approximations as follows: Let Yo = f(x) + g(x,0), Yn+1 f(x) + g(x,m), n 0 Show that, in a small neighbourhood of (0, 0), for example lxl -6, 11-5, in which the inequalities If(x)I 0/2, Ign(zw)1 1/2 hold, all the functions yn(x) satisfy ( (c) Hence, show that lyn(x)-ym(x)| m, and therefore Un(x) y(x) for lxl , where y(x) satisfies (**). Show the y(x) is unique, that is, if yi(x) is a second solution to (*), then y 1 (d)