# A strictly increasing convex function with zero right-derivative at a single point.

Let $$f: \mathbb{R} \to [0, \infty)$$ be strictly increasing and convex. Since $$f$$ is convex its right derivative at, say, zero exists and is finite.

Is it possible to construct an $$f$$ such that its right derivative is zero at some point?

Intuitively, this should not be possible since it would constitute a point where the tangent at $$f$$ (from the right) is flat and $$f$$ can not become smaller when we move to the left, i.e. we would either loose convexity or strictly increasing"ness". However, I cannot find a proper proof or counterexample.

Say $$D_Rf$$ is the right derivative. $$f$$ convex implies that $$D_Rf$$ is non-decreasing, while $$f$$ strictly increasing implies that $$D_Rf\ge0$$. So if $$D_Rf(0)=0$$ then $$D_Rf(x)=0$$ for all $$x<0$$, hence $$f$$ is not strictly increasing.