I know we should start with a convex function first, then use MVT to prove, but unable to figure out the entire proof.

(a) Let f : R > R be a differentiable function. Recall that the Mean Value Theorem states that for any distinct points a, b E R, f(a) f(b) ab : ff\") ' for some point c E ((1,6), Where f' is the derivative of f. Using the Mean Value Theorem, prove that the exponential function 9(95) 2: exp(a:) is convex over R. (b) Let f and g be convex functions on the reals. State Whether the following statements about f and g are true or false. If a statement is true, give a proof. If a statement is false, give both a counterexample and an example of functions for which the statement is true. (i) The sum f + g of the functions is convex. (ii) Suppose f and g are distinct functions. The difference f g is convex. (iii) Suppose 9 takes only non-negative values. The product f g is convex. (iv) The composition exp( f ) of the exponential function exp(3:) and f (It) is convex