ME 2352 Design Optimization Assignment TWO, due February 6th, 2024, 4:00 pm University of New Brunswick Department of Mechanical Engineering 1. By use of definition of linear dependency determine if: V = V = V3 = , V4 = [6.3. Winter 2024 a) V1, V2, V3 are linearly independent; and b) V1, V2, V4 are linearly independent. Show all your derivation to affirm your conclusion. 2. Using the Lagrangian function L(x) method, find points to satisfy the necessary condition for the following problem.: Maximize f(x1, x2) = Subject to x1 + x2 = 4 4x13x225x1x2 - 8x1 3. Quadratic Form and Matrix Definiteness: a) If possible, put functions and g in a vector/matrix quadratic form. b) Obtain the Hessian of functions f and g. c) By hand, use Sylvester's test to determine the definiteness of the Hessian for function f at points i, j, and k, and function g at points o, p, and q. - f(x1, x2, x3) = 2x + x12 + 2x2 + 3x 2x1x3 + 4 23 +7, X R g(x1,x2) = -2x3 + 5x2 7x1x2 + 4x2, XE R - i = [0,0,0], j = [4, 5,0], k = [4,-3,0] = [0, } }], p = [1.12, 0.878], q = [0.509, 0.181] 0 = d) For functions f and g, and for each of the points, what can you tell about the function at the specific locations? For instance, is the function convex, concave, or something else at those points? 4. The goal of this portion of the assignment is to create a MATLAB program that will determine the definiteness of a matrix. For this, you will need to Rev. Jan. 30th, 2024 Page 1 of 2 ME 2352 Design Optimization Assignment TWO, due February 6th, 2024, 4:00 pm University of New Brunswick Department of Mechanical Engineering Winter 2024 a) Create a single program that will take a matrix with the maximum size of 5x5 as input. The size() function can help you determine the dimensions of the input matrix. b) Verify the input is correct (e.g., matrix is square, etc.). c) Create a loop to check the sign of the determinant of the leading principal minors. d) Decide definiteness of the matrix based on the sign of the leading principal minors. MATLAB code should be submitted as the script (.m) file with at least one standalone function with one or more inputs and one or more outputs. There should be loops and, when appropriate, conditional statements. 5. Using the program created in Question FOUR, determine the definiteness of the following matrices. Verify your results using the eigenvalues of the matrices. 30 -32 26 -2 1 0 A = -32 37 -22 B = 1 -2 -2 , 26 -22 34 0 -2 -3 2 -6 C = 6 -5 5 9 145 500+ -6 -5 8 4 65272 -9- 8 7 7-6 429 9 1 7 -6 -6 12 -2 38 6 -2 9 -4 3 8 -4 ~ Yom D 4 F = DTD, G = -2 L-6 2020 -2 -6 10 9 32 9 14 -4] Submission Notes: 1. Submit hand worked solutions via the ME*2352 Dropbox located outside the ME office, including any plots of the results. 2. Submit your MATLAB code for Questions THREE and FOUR through D2L Brightspace. Thoroughly document your codes. 3. The MATLAB code should be submitted as the script (.m) file. 4. Specific instruction for the MATLAB code: a) State all your assumptions clearly and list all variables at the file with their corresponding values. top of your Rev. Jan. 30th, 2024 Page 2 of 2