ECO 3145: Spring/Summer 2022 Quiz 4
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ECO 3145: Spring/Summer 2022
Quiz 4
I) MCQ: True or False 3 marks=1 mark*3
Consider the problem ( for questions (1) and (2))
Max f(x1 , ..., xn ) subject to g(x1 , ..., xn ) ≤ b
xi eR
1-The complementary slackness condition [g(x) · b]λ = 0 means: either the constraint is binding, that is g(x) · b = 0 and λ ≥ 0, or the constraint is not binding and λ = 0.
2- If the Lagrangian function is concave with respect to the choice variables, then the KTCs are just necessary for a constrained maximum.
3- Consider the problem:
Max f(x1 , ..., xn ) subject to xi ≥ 0 for all i and gj (x1 , ..., xn ) ≤ cj for j = 1, ..., m
xi eR
To resolve this problem, we need m < n.
II) Consider the following integrals:
1 7
A = x (4x32 + 7x + 1) dx, B = x3 ′x dx
0 5
● Evaluate A, ,B, C and D.
III) Consider the problem:
=
dx, D = exp( ·2x + 1) dx
1,25 mark*4
Min · x2 y2 subject to · 2x · y ≥ ·2, x ≥ 0, and y ≥ 0
x,y
[You may use without proof the fact that ·x2 y2 is convex for x ≥ 0, and y ≥ 0]
a) Write down the KTCs.
b) Find the possible solutions for this problem.
c) Are KTCs sufficient for the optimum to exist? Justify your answer.
d) Find the optimal solutions for this problem.
2022-06-10