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Suppose that r is a double zero of the C2 function f, i.e., f(r) = f′(r)...

Suppose that r is a double zero of the C2 function f, i.e., f(r) = f′(r) = 0 but f′′(r) is not 0. Show that Newton’s method applied to f converges linearly with the asymptotic constant 1/2, i.e., show that

lim n->infinity | x(n+1)−r | / | x(n)−r | = 1/2.

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