No smoothness is assumed in this problem. (a) Let S C R be an interval. Let f:R S be convex, and let g:S + R be convex and monotone nondecreasing. Prove that the composition go f is convex on R. (b) Apply the result of part (a) to show that if f is convex on R and f(x) > 0 for all x E R, then fa is convex on R. (i) Give an example that this can fail to be true if we do not assume f(x) > 0 for all x E R. (c) Apply the result of part (a) to show that if f is concave on R and f(x) > 0 for all x E R, then is convex on R. (Hint: }=-=. (d) Suppose that f and g are convex functions on R. (i) Directly from the definition of convex function, prove that the pointwise max or upper envelope function h(x) = max{f(x), g(x)} is convex. (ii) Express the epigraph of h in terms of the epigraphs of f and g, and use this to give another proof that h is convex, using the fact that a function is convex if and only if its epigraph is convex