Let J be a closed interval of real numbers whose length is finite and positive. Let f be a real valued function defined on J which has a continuous second derivative at all points of J.

QUESTION: If P1,P2,P3 are any three pairwise distinct and non-collinear points on the graph of f(J), does there always exist at least one point p on J such that the absolute value of the curvature of this graph at the point (p,f(p)) is greater than or equal to 1/r-where r is the radius of the circle through the three points P1,P2,P3?