Question

# Bisection and Newton’s method.In this problem we consider the function f : [0, 10] → R defined by f(x) =

x2 − 1.

(a) Verify that f has a root ξ in the interval [0, 10]. Ahough this may be obvious by inspection, justify this by citing a theorem, and identify the root ξ.

(b) Show that in the interval [ξ − 1/2 , ξ + 1/2 ], the function f satisfies: |f”(x)| / |f'(y)| ≤ A ∀ x, y ∈ [ξ − 1/2 , ξ + 1/2 ]

Determine the smallest such value of A for the above statement to hold.

(c) Write down a sufficient condition on x0, the initial guess for Newton’s method, for quadratic convergence. Justify your answer by considering Theorem 1.8 (attached below) and using part (b).