Exercise 19, § 4

Let $f$ and $g$ be two finite real functions, $f$ being defined and continuous on an interval $I$, and $g$ defined and continuous on $\mathbf{R}$. Suppose that for every pair $(\lambda, \mu)$ of real numbers the function $g(f(x) + \lambda x + \mu)$ is convex on $I$. a) Show that $g$ is convex and monotone on $\mathbf{R}$. b) If $g$ is increasing (resp. decreasing) on $\mathbf{R}$, show that $f$ is convex (resp. concave) on $I$ (use prop. 7).