1. [10 points] (Convexity) Consider f: R R. For y,92 R", define the function g: R R by g(x) = f(x(y - y2) + y2), xR. (a) Prove that if f is convex on R", then g is convex on R for all y, y2 Rn. Hint: You can start your proof as follows. For any y, y2 R", a [0, 1], x, x2 R, g(ax + (1-a)x) = f((ax + (1 a)x)(y - y2) + y2) = f((ax + (1 - a)x) (y - y) + (a + (1 - a)) y) = f(a(x(y - y2) + y2) + (1 - a) (x2(y - y2) + y2)). Then use the convexity of f to continue the proof. (b) Prove that if g is convex on R for all y,92 R", then f is convex on R". Hint: You can start your proof as follows. For any y, y2 R", a = [0, 1], f(ay + (1 a)y2) = f(a(y - y2) + y2) = g(a) = g(a 1+ (1 - a).0). Then use the convexity of g to continue the proof.