Show that the set s xnbsp isin rn atx b is convex and

Question

X₁ + 3x₂ + 2x₃ + x₄ = 10

x₂+ x₃ + 2x₄ ≤ 8

3x₁ + x₂ + 2x₃ ≥ 6

x₁, X₂, X₃, X₄ ≥ 0, integer.

1. Show that the set S = {x  ∈ Rⁿ : a^Tx = b} is convex.

2. Show that the set S = {x E Rⁿ : aTx > b} is convex.

3. Let s₁, s₂ be convex sets. Show that S = S₁ ∩ s₂ is also convex.

4. Consider the functions f₁(x) = 10 - x and f₂ = 1/3x + 2, and let

g(x) = max{f₁(x)f₂(x)}.