Problem:
(Rootfinding and Optimization)
(a) Suppose that f is differentiable on [a, IN, Discuss how you might use a rootfinding method to identify a local extremum of f inside [a, b].
(b) Let f(x) = log x - cos x. Prove that f has a unique maxim= in the interval [3,4]. (NB: log means natural logarithm.)
(c) Approximate this local maximum using six iterations of the enclosure methods (Bisection and False Position) with starting interval [3,
(d) Approximate this local maximum using six iterations of the two fixed- point methods (Secant and Newton). For Newton's Method, use Po = 3. For the Secant Method, use po = 4 and 191 = 3.
(e) What is your best estimate for p, the location of the maximum?
(f) Provide the following two tables, comparing the four algorithms The headings for the two tables should be the following.
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Approximation pn
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Iteration n
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Bisection False Posn Secant Newton
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Iteration n
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Absolute Error IN - pl
Bisection False Pon Secant Newton
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(g) Plot the absolute error for all four methods on the same graph.
(h) What happens if you attempt to approximate the maximum by starting Newton's Method with po = 5?