Problem $2$

Consider the function $f(x)=x^3-6x^2+11x-5\mathrm{.}6.$

What theorem from basic Calculus makes it easy to check that $f$ has a root on the interval $0\mathrm{.}2,3?$ Explain and demonstrate.

Since we know $f$ has a root on the interval $0\mathrm{.}2,3,$ suppose we use Newton's method with initial guess right in the middle of the interval: $x_0=1\mathrm{.}6.$ Compute several iterations of Newton's method and discuss how well it is converging to a root.

What is another root$-$finding method we studied in this course that is slower but more reliable than Newton's Method here?

Write a computer program in MATLAB which first applies your slow$-$but$-$reliable$-$method several times and then uses Newton's Method. This program should have INPUTS: a function $f,$ the derivative of $f,$ bounds a and $b$ on an interval in which to search for the root, a maximum number of iterations $N,$ and an error tolerance TOL. The OUTPUT should be the approximation to a root of the function.

Use your function to find an approximation to the root in the example you gave above, with an error tolerance of ${10}^{-8}.$