3 (Operations that Preserves Convexity, 7.5 points each). Given f (x), f1(x), , fn(x) are convex function defined on Rn show that: (a) (Nonnegative weighted sum) If i 0 for all i = 1, 2, , n, then show that g(x) := n i=1 ifi(x) is a convex function. (b) (Pointwise maximum) g(x) := max i=1,...,n{f1(x), , fn(x)} is convex. (c) (Composition with an affine mapping) Given a matrix A Rnm and a vector b Rn, show g(x) := f (Ax + b) is convex with respect to x Rm. (d) (Restriction to a line) Fix some vector x, y Rn, define g : R R as g(t) := f (x + ty). Then g is convex with respect to t. 1