Compute the 'elbow' for a curve automatically and mathematically

Question

One example for curve is shown as below. The elbow point might be x=3 or 4. How to compute the elbow for a curve automatically and mathematically?

Answer 1

What you really want is the point with maximum curvature. When the slope is much smaller than 1, this can be approximated by the second derivative (as @ebo points out), but this is not always the case.