v Course Hero * *Course Hero X Final_Exam_3_Au X Unknown Gunme X G Graph Theory pa x | Access Bank: We x | Loading. x _ Loading.. X + X C @ File C:/Users/ALEX/Downloads/Final_Exam_3_Aug_2022.docx%20(1).pdf 9 Final_Exam_3_Aug_2022.docx (1).pdf 2 / 2 - 125% + Question 5 (40 points) Q.5) Multivariate functions: (a) Find all values for k so that f(x,y) = kx2 + 2ry + y? has a local minimum at (0, 0). Give your answer in the form of an interval. (b) Find the maximum and minimum of f(x, y) = y - x given that x2 + 2ry + 3y? = 3. (c) For the multivariable function f(I, y,2) = 212 + 51 32 + xy (i) Find the stationary points(s) of this function. (ii) Find the Hessian matrix. (iii) Find the eigenvalues and eigenvectors of the Hessian matrix at the stationary point(s). (iv) Classify the stationary point(s). (v) Orthonormalise the eigenvectors of the hessian matrix if they are not orthonormal. [8+7+(5+5+5+5+5) = 40 marks] Question 4 (15 points) Q.4) Probability, Bayes' Theorem: A computer antivirus system is designed such that the prob- ability of detecting the presence of a virus attached to an email is 98%. However if no virus is present, it still reports (falsely) that a virus is present with a probability of 5%. At any time, the probability that a virus is present is 7%. (a) What is the probability that no virus is present given that a virus is detected? (b) What is the probability that a virus is present given that the virus is detected? (c) What is the probability of misclassification? [5+5+5 =15 marks] 1 89OF Partly sunny Q Search O 2:11 PM 5/30/2024X | G GraphTheorypart1 X | @ AccessBankWelcor X | B HomeworkHelp-0f X v Course Hero X Course Hero X @ Finallbem_3Aug2l X I} Unknown Gunmen c @ File C:/Users/ALEX/Downloads/Final_Exam 3 Aug 2022.docx%20(1).pdf Final_Exam_3_Aug_2022.docx (1).pdf 105% =+ = OD Question 1 (15 points) Q.1) Graph Theory: Consider the following incidence matrix of a graph G = (V,E) with V = {a,b,e,d, e} and {e1, 2, 3, 4, 3, 6, 7, 5, 9, 10, 11, 12} Based on the information you obtain from the incidence matrix M, answer these questions; (a) Draw the graph (b) Find the adjacency matrix of this graph (c) Determine the type of the graph. How many paths of length 2 are there between nodes and b (without direct enumeration)? (d) (e) Consider A;, i = 1,...,5 are eigenvalues of the adjacency matrix A. Without finding the eigenvalues of the adjacency matrix, determine the values of 3 A; and 3 A2, Justify your i=1 i=1 answer. [3+3+3+3+3=15 marks] Question 2 (15 points) Q.2) Linear Algebra: Based on the different values of o T-2yz=a 2r+2y+br=2 (a) Discuss the states of the system (has a unique solution, many solutions, and no solution) using Gaussian elimination. (b) Considering the representation of the system as Ax = b, find the rank and the determinant of the matrix A when b =1 () Find the nullspace of A, namely, N(A), when b= 1 218PM & Q Search & I.D@.?@' G@o @E iy DB 00 B 9 @) 89F Partly sunny