This exercise considers solving ( ) = 0, where                                 This function is shown in Figure 2.24. (a) Show that  ( ) > 0 for all . (b) Describe what happens if one uses Newton’s...

This exercise considers solving
() = 0, where

This function is shown in Figure 2.24.

(a) Show that
 () > 0 for all.

(b) Describe what happens if one uses Newton’s method with

₀
= 1. Also, explain why essentially the same thing happens if you use

₀
= −1.

(c) Experiment with Newton’s method, and find the largest (positive) value of

₀
that will result in Newton’s method converging to the correct solution. You only need to find x0 to two significant digits. Also, give the corresponding value of

₁. Note that keeping track of what happens to

₁
will be helpful for part (d).

(d) Explain why the value of

₀
you found in part (c) can be found by finding the positive solution of 2 () =
(). What exactly is the relationship between

₀
and

₁
that gives rise to this equation?