Question

Problem 2. Determine the critical point(s) of T1 T3 and classify the values of F at the critical points. Show all of your work. As part of your answer, check that the characteristic polynomial of the Hessian at the critical point(s) is equal to (A+ 1)(A2-A-2π, (This factorization will help you to find the eigenvalues of the Hessian.) [6 marks

Answer 1

Given that F[x_1 x_2 x_3] = pi x_1 x_2 + x_1 x_3 + pi x_2 x_3 Let us find out nabla f(x) therefore nabla f(x) = (F_x_1, F_x_2, F_x_3) = 0 = (pi x_2 + x_3, pix_2 + pix_3, x_1 + pix_2) = 0 implies pi x_2 + x_3 = 0 - 1 pi x_2 + pi x_3 = 0 - 2 x_1 + pix_2 = 0 - 3 consider [0 0 1 pi 1 pi 1 1 0][x_1 x_2 x_3] = [0 0 0] As det [0 0 1 pi 1 pi 1 1 0] - 0 - pi(-1) + 1(-1) = pi - 1 notequalto 1 implies [x_1 x_2 x_3] = A^-1 [0 0 0] = [0 0 0] therefore (x_1, x_2, x_3) = (0, 0, 0) is the only critical point Now H = [F_x_1x_1 F_x_2x_1 F_x_3x_1 F_x_1x_2 F_x_2x_2 F_x_3x_2 F_x_1x_3 F_x_2x_3 F_x_3x_3] = [0 pi 1 0 pi pi 1 pi 0] char(H) = \r\n < null >
hessian matrix and nature of critical points