Using python 3 apply Newtons Method to find both roots of the function f (x) =14xe^(x2) 12e^(x2) 7x^3 + 20x^2 26x + 12 on the interval [0,3]. For each root, print out the sequence of iterates, the errors ei, and the relevant error ratio ei+1/e2 i or ei+1/ei that converges to a nonzero limit. Match the limit with the expected value M from Theorem 1.11 or S from Theorem 1.12.

a Case, as su THEOREM 1.12 Assume that the (m + 1)-times continuously differentiable function f on [a, b] has a mul- tiplicity m root at r. Then Newton's Method is locally convergent to r, and the error e; at step i satisfies ei+1 lim 100 ei = S, (1.29) where S= (m - 1)/m. THEOREM 1.11 Let f be twice continuously differentiable and f(r) = 0. If f'(r) #0, then Newton's Method is locally and quadratically convergent to r. The error e; at step i satisfies ei+1 = M, lim where M= f"(r) 2 f'(r)