Chapter 4.8, Problem 6E

The function $f(x)=\frac{4x}{x^2+4}$ is differentiable and has a local maximum at x = 2, where $f^{\prime }\left(2\right)=0$ (see figure).

  a.    Use a graphical explanation to show that Newton’s method applied to $f(x)=0$ fails to produce a value x₁ for the initial approximation x₀ = 2.

  b.    Use the formula for Newton’s method to show that Newton’s method fails to produce a value x₁ with the initial approximation x₀ – 2.