Consider the problem of minimizing f and g using the Newton method (NM) with y = 1. (a) f(x) : = a + bTx + x™Cx, where a E R, b E R", and C > 0. (i) What is the minimizer x* of f? Is it unique? (ii) Show that NM converges to x* in just one iteration from any initialization x° e R". (b) f(x) = (x/a) – log(x), for a, x > 0, where log denotes the natural logarithm. (i) What is the minimizer x* of g? Is it unique? 1 and x° = 1/2? 1 and x° (ii) What are the values of the first 3 iterates generated by NM for a = (iii) What are the values of the first 3 iterates generated by NM for a = = 10? (iv) Find an interval I C R such that for a > 0 and x° e I, NM monotonically converges to x*.

Suppose f is differentiable at a and f(a) # 0. (a) Prove |f| is differentiable at a by using that |f| = Vf2. What are the possible values of |fl'(a)? (b) Give a counterexample that shows the conclusion need not hold if f(a) = 0. Identify where your calculations for (a) fail in this case.

Please show work for Calculus 1

The function f(x) = x3 + 5x – 4 has a single zero near x0 = 1. Use the backward difference method to estimate f’(1) with a step size of 0.01  and use this value in an approximate version of the Newton-Raphson method to derive one improvement on x0. What is the root x1 of the function after one improvement?

Use Newton's Method to approximate the zeros for f(x), and be sure to continue the process until two successive iterations differ by no more than 0.001, which is your delta-x value. You should find three (3) answers between x = - 4 and x = 4. Please round your answers to four (4) decimal places. f(x) = 2x³ - 8x² + 3x +5