A function f : Rn R is called weakly increasing if f(x) f(y) whenever x y. It is called strictly increasing if f(x) > f(y) whenever x y. Finally, f is called strongly increasing if f(x) > f(y) whenever x y, with x = y. Identify which property is satisfied in each of the following cases. Since the problem is worth 2 marks, choose four out of the six cases.

(a) Let f(x1, x2) = c1x1 + c2x2 defined on R2 for c1, c2 > 0.

(b) Let g(x1, x2) = ln x1 + ln x2 defined on R2 ++.

(c) Let d0 : R2 R+ be the Euclidean distance to 0.

(d) Let dz : Rn R+ be the Euclidean distance to z. (e) Let u(x1, x2) = x1 1 x2 2 defined on R2 ++, with i > 0 for i = 1, 2. (f) Let v(x1, x2) = min{x1, x2} defined on R2 ++.