Regarding the following theorem: If $f$ is infinitely differentiable on an interval $I$, and $f^{(n)}(x)\ge0$ for all $n\in\mathbb N$ and $x\in I$, then $f$ is analytic on $I$.
This theorem is proven for non negative derivatives. My question: what if some or all of the derivatives of the function are negatives ? Does that imply that the function is not analytical, and why ?