Let % be a binary relation defined on RL + that is complete and transitive. Let x,y ∈ RL + be consumption vectors. Here are some important definitions: • The relation % is convex if for all x and y such that x % y, αx + (1−α)y % y for all α ∈ [0,1].2 • The function u: RL + → R is concave if for all x,y ∈ RL +, and α ∈ [0,1], u(αx + (1−α)y) ≥ αu(x) +(1 −α)u(y). • The function u: RL + → R is quasi-concave if for all x,y ∈ RL +, and α ∈ [0,1], u(αx + (1 − α)y) ≥min {u(x),u(y)}.

求证明：Suppose there are two goods, i.e. L = 2. Consider the utility function u(x) = x2 1x2 2. Verify that this function is not concave. Verify that it is quasi-concave.