[2 pts] Let f R R be differentiable. We say that f is convex if for all x, y R" and all t = [0, 1], f(tx+(1-t)y) tf(x) + (1-t)f(y) i.e. that the graph of f is below its chords. (a) [1 pt] Prove that f is convex if and only if f(y) f(x) Vf(x) (y-x) for all x, y R", i.e. the graph of f is above its tangent lines. (b) [0.5 pt] Assume that f is C2. Prove that if f is convex then n , ()hh; 2 for all x, h Rn. i,j=1 We recall that for a function g: RR twice differentiable, we have the following Taylor formula: for all a, b ER, there exists c E [a, b] such that g(a) = g(b)+g' (b)(a - b) + g"(c) 2 (This is no longer true for functions of several variables.) (c) [0.5 pt] Applying the previous formula to the function g(t) = f(x+ th) between a = 0 and b = 1, prove the reverse implication in the