1. Let f (x ) := |x |p for all real x . For what values of p is it true that (a) f is convex; (b) the derivative f I(x ) exists for all x ; (c) the second derivative f II exists for all x?
2. Let f be a convex function defined on a convex set A ⊂ Rk . For some fixed x and y in Rk let g(t ) := f (x + ty) + f (x - ty) whenever this is defined. Show that g is a nondecreasing function defined on an interval (possibly empty) in R.