Is a smooth, non-constant, continuous function with more than one critical point necessarily not convex?
I figure if you have two critical points, you have an inflection point between by Rolle's Theorem. If that inflection point isn't also a critical point, it implies a sign change of the second derivative, which means it switches from convex to concave or vice versa.
So id conclude convex shape implies at most one Critical point, and so a unique minimum or maximum ignoring boundaries of the domain.
After some research I think I've found an answer I'm satisfied with.
Concavity can change at inflection points. The second derivative being zero and the next higher order, non-zero derivative being an odd power implies an inflection point.
Critical points and inflection points may not be such under Euclidean Transformations of the curve.
Curvature and its derivative remain unchanged under such rotations and give you information about concavity that holds after such transformations.