Critical points and Convexity? Function $f(x)$ has no critical points in $M$, can we say $f(x)$ is either convex or concave over $M$? 

If $f(x) = 2x+\sin(x), x\in \mathbb{R}$, then $f^\prime(x) = 2+\cos(x)$ is nowhere $=0$, hence there is no critical point. However, $f^{\prime\prime}(x)=-\sin(x)$ changes sign, so $f$ is neither convex nor concave. By scaling you can do that on any interval, as small as you like.
