SPRING TRIMESTER MIDTERM EXAMINATION - 2020/2021
MATH2003J Optimisation in Economics
1. (Full mark of Question 1: 15 marks)
(a) Determine whether each of the following statements is True or False.
No explanation is needed when answering 1(a)(i) to 1(a)(iv).
(i) Let p be a critical point of a twice differentiable function f : Rn → R.
If ∆k (p) > 0 for all k = 1,... ,n, then f has a local maximum at p. [2]
(ii) Let p be a critical point of a twice differentiable function f : Rn → R.
If ∆k (p) ≠ 0 for all k = 1,... ,n, then f has a saddle point at p. [2]
(iii) The Hessian matrix of the function f(x,y) = x2 − 6xy is
[2]
(iv) The following is not a linear programming problem in standard form.
Maximize z = x1 − 2x2 + x3
subject to x1 + 3x2 + x3 ≥ −6
6x1 + x2 − 7x3 ≤ 3
x1, x2, x3 ≥ 0 [2]
(b) Determine whether the following statement is True or False.
Justify your answer.
Let f : R2 → R be a linear function in x and y. There are infinitely many solutions to the linear programming problem:
Maximize z = f(x,y)
subject to 5x + 2y ≥ 10
3x + y ≤ 3
x,y ≥ 0. [7]
2. (Full mark of Question 2: 42 marks)
(a) Find and classify all critical point(s) of the function
f(x, y, z) = x2 − xy + y2 − 3yz + 7z2 + 6. [20]
(b) Use the method of Lagrange multipliers to find the maximum and the mini-mum of the function f(x,y)=8x2 − 8y2 + 5 subject to the constraint
4x2 + y2 = 4. [22]
3. (Full mark of Question 3: 43 marks)
(a) Solve the following linear programming problem by the simplex method:
Maximize z = 12x1 + 10x2
subject to 2x1 + 3x2 ≤ 120
9x1 + 6x2 ≤ 390
x1, x2 ≥ 0. [21]
(b) Solve the following linear programming problem by the simplex method:
Maximize z = 2x1 + x2 + 3x3
subject to x1 + x2 + x3 ≥ 10
−x1 − x2 − x3 ≥ −20
x1, x2, x3 ≥ 0. [22]