1. multiple minima (5 pts.). We will explore the behavior of minimization algorithms for a simple univariate case with multiple minima. Let L(x) = 24 - 14x3 + 60x2 -70r. (a) Plot the loss function L over the set [-1, 8). Is it convex? If yes, why? If no, are there regions where it is (locally) convex? (b) What do you expect the behavior of Nelder-Mead or Newton's method will be? (c) Use the Nelder-Mead algorithm to find the argument that minimizes L. Initialize your two points randomly in [0, 2). Plot a vertical bar where the minimizer is, together with the function. Compare to the result of R's optim just for sanity check. (d) Keep track of the "best" Nealder-Mead point at each iteration (21 ) and make two plots showing this sequence. In the first plot show the values L(a, ) for different iterations on top of the loss function L. In the second one plot the convergence behavior as in the plot of page 3, w07dl notes. (e) Repeat the above two problems with initial two points in [6, 8). Explain your result. 2. Newton's method (5 pts.): Perform a similar analysis as in question 1 but using Newton's method. Remember to keep track of your iterates and plot their convergence behavior. Do it for both an initial value somewhere in [0, 2] and somewhere in [6, 8]