Chapter 3.8, Problem 21E

Using Newton's Method Consider the function

$f(x)=x^3-3x^2+3$

(a) Use a graphing utility to graph f.

(b) Use Newton’s Method to approximate a zero with $x_1=1$ as the initial guess.

(c) Repeat part (b) using $x_1=\frac{1}{4}$ as the initial guess and observe that the result is different.

(d) To understand why the results in parts (b) and (c) are different, sketch the tangent lines to the graph of f at the points $(1,f(1))$ and $\left(\frac{1}{4},f\left(\frac{1}{4}\right)\right)$

Describe why it is important to select the initial guess carefully.