Prove that () is a convex function.

Proposition 5.C.1: Suppose that (. ) is the profit function of the production set Y and that y( .) is the associated supply correspondence. Assume also that Y is closed and satisfies the free disposal property. Then (i) x( .) is homogeneous of degree one. (ii) x(.) is convex. (iii) If Y is convex, then Y = { ye R': p.y S n(p) for all p > 0}. (iv) y( .) is homogeneous of degree zero. (v) If Y is convex, then y(p) is a convex set for all p. Moreover, if Y is strictly convex, then y(p) is single-valued (if nonempty). (vi) (Hotelling's lemma) If y(p) consists of a single point, then n(.) is differentiable at p and Vn(p) = y(p). (vii) If y( .) is a function differentiable at p, then Dy(p) = D2x(p) is a symmetric and positive semidefinite matrix with Dy(p ) p = 0. Properties (ii), (iii), (vi), and (vii) are the nontrivial ones. Exercise 5.C.2: Prove that x( . ) is a convex function [Property (ii) of Proposition 5.C.1]. [Hint: Suppose that ye yap + (1 - x)p'). Then nap + (1 - a)p') = ap.y+ (1 - a)p'.y s an(p) + (1 - a)n(p').]