Find the maximum of f (x, y) = 4x + 2y + x2 − 2x4 + 2xy − 3y2 using the steepest ascent method with initial guess x = 0 and y = 0. (a) (5 points) find the gradient vector and hessian matrix for the function f (x, y). (b) (5 points) find the direction of steepest ascent at (0,0). the direction found in this step will be our search direction in the next iteration. (c) (10 points) find the point that maximizes the function f along the search direction. 2.

In a unit cell of fcc structure, indicate the position of one of the tetrahedral interstitial sites by plotting the corresponding tetrahedron. do the same for one of the octahedral interstitial sites (in a separate fcc unit cell). mark the positions of all the tetrahedral and octahedral interstitial sites in the unit cells. how many equivalent tetrahedral sites and how many equivalent octahedral sites are contained in a fcc unit cell, respectively. what is the coordination number for a tetrahedral site and what is the coordination number for an octahedral site?