Chapter 4.9, Problem 436E

Newton’s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton's method to the derivative function $f\text{'}\left(x\right)$ to find

its roots, instead of the original function. For the following exercises, consider the formulation of the method.

436. To find candidates for maxima and minima, we need to find the critical points $f\text{'}\left(x\right)=0$ . Show that to solve for the critical points of a function $f\left(x\right)$ , Newton's method is given by $x_{n+1}=x_n-\frac{f\text{'}\left(x_n\right)}{f"\left(x_n\right)}$