Answer Happy

This problem sheet asks you to prove some well known results. Although the algebra is easy the proofs are not entirely straightforward. There are marks assigned to the readability of the solution and also how well laid out and explained the steps you make are. (A good proof needs to be easy to follow: you need not comment on trivial algebra, but there should not be steps that are difficult to follow). 1 (a) Starting from the definition of a convex function where, for a € (0,1), f(ax + (1 - a)y)<af(x) + (1 - a) f(y) Let a = €/(x - y) and rearrange the inequality to give (1) (C-y) (5(4+0) – 49) on the left-hand side. Taking the limite + O show that the function f(x) lies above the tangent line t(x) = f(y) + (x - y) f'(y) going through the point y. (b) Sketch the tangent line, t(2), at the point y in the graph shown below. f(x) у (c) Starting from the inequality for a convex function, f, f(x) > f(y) + (x - y).f'(y) (2) consider the case y = x + €, then by Taylor expanding f(x + e) and f'(x + €) around x and keeping all terms up to order , show that f'(2) > 0. (d) Prove that is convex.