## Exercises

1.
Consider f(x) = x4 - 16x - 12 and let r = 32. First let x0 be 1.587. What root is found? Next let x0 be 1.588. What root is found? Is a root really found? Why do two initial guesses so close together lead to different roots? Use your knowledge of the explanation of how Newton-Raphson works to answer this part of the problem.
2.
Use Newton-Raphson to find all points at which the graph of the function
f(x) = x5 + 3x4 - 17x3 + x2 - 8x + 4 has a horizontal tangent line. Explain why you can be sure that you have indeed found all such points.

3.
Consider , let x0 = 2 and r = 30. Discuss what happens. Explain why this behavior occurs. What is the signficance of the numbers that appear? Find all numbers which when used as x0 will lead to a root. Hand in appropriate printouts. Again note that Newton-Raphson can be very sensitive to the initial guess that is used.

4.
Let f(x) = 3x4 - 4x3 - 84x2 + 288 x + 732. Let x0 = 2 and r = 10. Run Newton. What causes the result you get to occur? Explain this in terms of the Newton-Raphson process.

5.
Construct a polynomial function g(x) of degree 5 or greater that has no real roots. Explain why you are sure it has no real roots. Let r = 20. Select a value for x0 and run Newton. Try a different value for x0 (one not close to the original value) and run Newton. Discuss the output for these two runs. Describe what happens.