QUESTION 4 a) For matrix A, determine the associated quadratic form f(x1, x2) = x²Ax. A = 1- (11) 3 [3 marks] b) Determine if A is a positive definite or a negative definite matrix and hence, de- termine the nature of the stationary point of the associated quadratic f(x1, x2). [3 marks] c) The quadratic f(x1, x2) can be used to model one section of a prospective building site. The best potential site is close to a road and part of that road can be modelled using the constraint: g(x1, x2)=3x14x24=0 i) Form a Lagrangian function L(x1, x2, y) with y as a Lagrangian multiplier and determine the partial derivatives OL ƏL მე მე OL and მყა [5 marks] ii) Form a system of linear equations. Express the system as an augmented matrix and determine x1, x2 and y. Show all steps of the solution. [6 marks] iii) The altitude [in metres] of the site above a flooding plain is given by A[m] f(x1, x2) 8. The site at this location must be built at an altitude greater than 4 m above the flooding plain. Does the location satisfy this rule? [Total marks: 20] [3 marks] Semester 2 2023-2024 KL5002: Further Mathematics Page 5 of 8