Sppose a consumer has a utility functionnbspu x1nbsp x2

Question

Suppose a consumer has a utility function: u( x1 , x2) = min {x1, 2x2}, 

i.) Prove that the set of bundles ( x1, x2) = min {x1, 2x2} ≥ 10 is a convex set. 

ii.) Prove that, for any bundles (x1, x₂) and (y₁, y₂), the utility function satisfies:

u( x1/2 + y₁/2 ,  x2/2 +y₂/2 ) ≥ min { u(x1, x2) , u(y₁ , y₂)}