Newton's method

$($a$)$ Implement in code Newton's method to solve the following problem:

$f(x)=e^{6x}+1\mathrm{.}441e^{2x}-2\mathrm{.}079e^{4x}-0\mathrm{.}333=0,$xin$[-5,5].$

with $|f(x)|$ ftol $={10}^{-12}.$

$($b$)$ Run your code using the following $3$ initial guesses: $x_0=-5,2,5.$ Summarize your numerical results.

$($c$)$ Numerically estimate the convergence rate for each of your computer runs. Analyze the results and

discuss the advantages and disadvantages of this algorithm.