# Rootfinding and Optimization : Newton's, Secant, Bisection, False Position Methods

3. (Rootfinding and Optimization)

(a) Suppose that f is differentiable on [a, b]. Discuss how you might use a rootfinding method to identify a local extremum of f inside [a, b].

(b) Let f(x) = logx ? cosx. Prove that f has a unique maximum 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, 4].

(d) Approximate this local maximum using six iterations of the two fixed-point methods (Secant and Newton). For Newton's Method, use = 3. For the Secant Method, use P0 = 4 and p' = 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 headin for the two tables should be the following.

[TABLE]

(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 p = 5?

Please see the attached file for the fully formatted problems.

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#### Solution Summary

Roots are found and optimized using Newton's, Secant, Bisection, False Position Methods. Methods are compared. The solution is detailed and well presented.