Consider the function $f(x)=2e^{x-1}-x^2-1.$

$($a$)$ Write out the analytic formula for Newton's method applied to this $f(x).$

$($b$)$ Write code to apply Newton's Method to find the only root to as much accuracy as possible.

For example, you could stop the iteration when $|f(x_i)|0\mathrm{.}5{10}^{-15}.$ Report the number of

iterations taken and the forward and backward errors of the best approximation. Use the initial

guess $x_0=0.$ A template to help you get started is provided in the file Homework$2.$mlx$.$

$($c$)$ Find the root's multiplicity.