Note: N = last digit of your student number Question 1 (25 Marks) Use the simplex method to solve the following problem: Maximise p=3x+4y+2z subject to the constraints 2x + 2y + 4z ≤6 2x+5y+ (N + 2)z ≤7 5x+(14 N)y + 2z ≤ 8 - x,y,z ≥0 Question 2 (25 Marks) Use the Lagrange multiplier method to find the dimensions of the box (Fig 1) with the lowest surface area. The box has top face missing and its volume must be at least 10 + N cm³. Fig. 1. A rectangular box with a top face missing. Question 3 (25 Marks) Use the golden section search method to solve the following problem: Maximize f(x)=-10x² - (30 + 4N)x + 10 over the interval 1≤x≤ 10. Locate the optimum value of x within an interval of uncertainty whose length is at most 15% of the initial search interval. Present your solution in a tabular format. 3 of 3 Use the golden section search method to solve the following problem: Maximize f(x)=-10x2 (30+ 4N)x + 10 over the interval 1 ≤ x ≤ 10. Locate the optimum value of x within an interval of uncertainty whose length is at most 15% of the initial search interval. Present your solution in a tabular format. Question 4 (25 Marks) Use the method of steepest descent to find the maximum of the following objective function: 2 U(x, y) = - (+1)(x- N N +1)(x+N)x(+1)(y-N)y + 40 xy 20-N 10 Start the search at the location (x, y) = (11N, 1+ N) and stop when [AU] < 1 or after 6 iterations. Use step length Ax equal to 0.2.

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Note: N = last digit of your student number Question 1 (25 Marks) Use the simplex method to solve the following problem: Maximise p=3x+4y+2z subject to the constraints 2x + 2y + 4z ≤6 2x+5y+ (N + 2)z ≤7 5x+(14 N)y + 2z ≤ 8 - x,y,z ≥0 Question 2 (25 Marks) Use the Lagrange multiplier method to find the dimensions of th