Let's call function f(a) nice if it is defined everywhere on 1, 1 10 (a) 10, the function is infinitely differentiable on t1, 11, and has a single minimum on (-1, 1) (trivial case of extrema at the ends are excluded). For example, f(r) is a nice function, but sin (1/a2) or z' are not. Suppose you use a numerical procedure to find the minimum z of a nice function. We call such numerical solution right if laappror zeractl 10 3, aeract being the exact answer. Otherwise, the solution is called urong. Note that our definition is very generous: generically, one expects the correct solution to be within ve of the exact, that is within N 10 Use Mathematica to explore the straightforward Newton's method for finding local minimum Nice functions have only one minimum by definition, so not to worry. Use FindMinimum l; let Mathematica select all input parameters automatically, except the method "Newton", which you specify explicitly (see examples on the class site, you may use any of them as a template). Explore a nice function f(a) ar 2 bacA, consider limiting cases such as a -0, 0, and some intermediates. Present convergence graphs (use FindMinimumPlot Make your conclusions Now that you understand well how Newton's method works, show it! ome up with a nice function use the definition above) that breaks Newton's method that is Mathematica, with default settings, gives a wrong solution (see above defs.) without so much as a peep no warnings or errors Present convergence graphs (use FindMinimumPlot D) Explain the failure Reason which method from your C&K textbook may mitigate the problem. Give an intuitive explanation for why the alternative to Newton's works; in a few sentences, give at least one pro and one con for your alternative See if you can harness Mathematica's unique functionality and options to make it find the right solution, using the same Newton's. In fact, you may be able to get to it within Find what machine epsilon is for your machine. Use code on the class site). Explain why the solution, while useful in research, is not very useful in situations when you need to quickly find lots of minima as part of a larger code written in a standard language such as C