Determine whether convex or not:

Determine whether the following functions are convex or not: (a) dom(f) = {r ER' | x1 > 0, x2 > 0} and 1 f ($1, X2) = X1 2 (b) dom(f) = {r ER? | x] ER, 2 > 0} and f(x1, X2) = X 2 (c) dom(f) = {x E R" | (a, x) + b > 0}, where a E R" and b E R are given with a * 0, and 1 f (21, . . . , In) = (a , x ) + b (d) Let any denote the ith largest component of a vector x E R". Let r be a given integer satisfying 1 oxi =1 Hint: You can use the fact that the function gr(x) = _ _1 Xy] is convex for every k = 1, 2, ..., n (shown in the class)