(i) 10 point Suppose that f is twice differentiable on [a, b] and that there is a number p = [a, b] such that f(p) = 0.

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I 10 Point Suppose That F Is Twice Differentiable On A B And That There Is A Number P A B Such That F P 0 1 (277.11 KiB) Viewed 25 times

(i) 10 point Suppose that f is twice differentiable on [a, b] and that there is a number p = [a, b] such that f(p) = 0. Assume that f'(p) ‡0. Show that there is S> 0 such that the sequence {pk} defined by Pk+1 = Pk f(pk) f'(pk)' k > 1 converges to p for any initial condition po € [p−8, p+8]. (Hint: A key step is to show that you can apply the fixed point theorem to the above iteration for some > 0.) Solution See the slides (ii) 10 point Consider the following function f(x) = x² + 7x + 12 Construct the MATLAB code of the iteration in (i). Also, find the roots of f via the iteration in (i) when the initial conditions are po = −3.8 and Po = -2.8.