Prove the special case of Theorem 4.2.5 in which the function g is twice continuously differentiable and X is one-dimensional. You may assume that a twice continuously differentiable convex function has nonnegative second derivative. Expand g(X) around its mean using Taylor’s theorem with remainder. Taylor’s theorem with remainder says that if g(x) has two continuous derivatives g' and g'' at x = x0, then there exists y between x0 and x such that g(x) = g(x0) + (x − x0)g'(x0) + (x − x0)2/2g'' (y).