Show that the equation $^3 + 2 = 5$ has exactly one solution on $[1, 2]$. Then find the solution correct to $4$ decimal places using Newton’s method.
Looking at the problem, I wonder how do I finish this problem without using the initial guess, but I can't seem to figure it out. I have absolutely no idea on how to finish this problem.
Take the derivative to show the function is increasing monotonically. The note the function starts negative at the beginning and reaches a positive value at the end. So by the Intermediate Value Theorem, it must hit zero somewhere on the interval. It can do this only once since it is increasing.
From there you can use Newtons Method or the cubic formula. 1 is probably a good starting point.