convexity and first derivative

Question

Let $\phi$ be a differentiable function on an interval $(a,b)\subset R^1$.

If $\phi '$ is non-decreasing, then $\phi$ is convex.

But, is the converse true?

Does the convexity of $\phi$ necessarily imply that $\phi '$ is non-decreasing?

Answer 1

Yes,

$\phi(x)$ is convex if and only if $ \nabla_x^2\phi(x) >0$, which is means that $\nabla_x\phi(x)$ be non decreasing.