Determine a polynomial function fx that has alpha as its

Question

Problem 1:

The true root of f (x) = ⁴√x is obviously α = 0. Consider trying to ?nd this root using Newton's method.

(a) Give the iteration formula for Newton's method. Give as simpli?ed a form as possible.

(b) Show that for any x₀ ≠ 0, Newton's method always fails.

Problem 2:

Suppose we wish to determine the number α = √15.

(a) Determine a polynomial function f (x) that has α as its true root.

(b) Find the iteration formula for the Secant method.

(c) Let x₀ = 4 and x₁ = 5 and compute the next two iterates x₂ and x₃.