代做MATH2003J Optimisation in Economics SPRING TRIMESTER MIDTERM EXAMINATION - 2020/2021代写Processing

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]