I have function $f(x)$ that starts as concave up and becomes concave down after some point (the "threshold").
What's the mathematically rigorous way to quantify the sharpness of the "threshold"? I think it is safe to define as a threshold any point where the direction of concavity changes (second deviation of $f$ becomes zero and third deviation is not zero, if memory serves), but I don't know how to capture the "sharpness".
What you refer to as a "threshold" is generally called a point of inflection. Indeed, a point of inflection is defined as any point where a function changes concavity (i.e. where the second derivative of the function changes sign). Now, as for characterizing the sharpness, possibly the easiest way would be to find the third derivative of the function, which would tell you how quickly the second derivative is changing over the interval you are looking for.