Let $f: \mathbb R \rightarrow\mathbb R $ be twice continuously differentiable. Suppose further that $f$ is bounded and $f’’(x)\geq 0$ for every $x \in \mathbb R $. Then prove that $f$ is infinitely differentiable.
Except constant functions I am not able to get any function satisfying the hypothesis. The exponential function satisfies all properties but is unbounded. $\arctan$ satisfies all properties except that its second derivative is not nonnegative in $\mathbb R$. Are there non-constant functions satisfying the hypothesis? How do we prove the result?