Suppose that I is a closed interval and x0 I. Suppose further that f is differentiable on

Suppose that I is a closed interval and x0 ˆˆ I. Suppose further that f is differentiable on R, that f'(a) ‰  0 for some a ˆˆ R, that the function

satisfies F(I) Š‚, and that there is a number 0 a) Prove that |F(x) - F(y)| b) If xn:= F(xn-1) for n ˆˆ N, prove that |xn+1 - xn| c) If xn = xn-1 - f(xn-1)/f'(a) for n ˆˆ N, prove that

exists, belongs to I, and is a root of f; that is, that f(b) = 0.

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