[ntownsend]

No, in this case, a convex function is one where you can draw a straight line between any two points on its graph and not have the line intersect the graph at a third point.

[pieceofpeace]

You are not contradicting what fgimenez said, its the same condition stated differently.

edit: if two statements are not essentially same (and I am wrong), could you please explain it.

[jamii]

The logarithm function is convex by this definition but has no maximum/minimum point and never has 0 derivative.