OIMU 2005 Problem 6

A smooth function $f:I\to \mathbb{R}$ is said to be totally convex if $(-1)^k f^{(k)}(t) > 0$ for all $t\in I$ and every integer $k>0$ (here $I$ is an open interval).

Prove that every totally convex function $f:(0,+\infty)\to \mathbb{R}$ is real analytic.Note: A function $f:I\to \mathbb{R}$ is said to be smooth if for every positive integer $k$ the derivative of order $k$ of $f$ is well defined and continuous over $\mathbb{R}$ . A smooth function $f:I\to \mathbb{R}$ is said to be real analytic if for every $t\in I$ there exists $\epsilon> 0$ such that for all real numbers $h$ with $|h|<\epsilon$ the Taylor series $$\sum_{k\geq 0}\frac{f^{(k)}(t)}{k!}h^k$$ converges and is equal to $f(t+h)$ .