MATH2003J, OPTIMIZATION IN ECONOMICS,
BDIC 2023/2024, SPRING
Problem Sheet 11
Θ Question 1:
Consider the sets:
A = {(x, y) ∈ R2 ∣ x2 + y2 ≤ 4, ∣y∣ ≤ 1}
B = {(x, 0) ∈ R2 ∣ x ∈ [0, 10]}
C = {(x, y) ∈ R2 ∣ y2 ≤ x − 4, x ≤ 8}
(a). Sketch the set A U B U C and decide if it is closed and bounded.
(b). Are A, B , C convex? Is the set A U B U C convex? Justify your answer.
(c). Consider the function f : A U B U C → R defined by f(x, y) = x + y. Find the absolute maximum and minimum of f.
Question 2:
Decide if the following sets are convex or not, and justify your answers:
(1) A = {(x, y) ∣ x + y ≤ 10, x ≥ 0, y ≥ 0}
(2) B = {(x, y) ∣ x2 + y2 ≤ 4}
(3) C = Z ⊂ R.
(4) D = {sin(1/x) ∣ x ∈ (0, +∞)} ⊂ R.
(5) E = {(x, y) ∈ R2 ∣ y2 ≤ x}.