Can you determine the average second derivative from a set of points? Let us say we have a smooth function $f$. We can find the exact average of $f'$ on the interval $[a,b]$ via

$$\bar{f'}=\frac{f(b)-f(a)}{b-a}$$

My question is, can you find the exact average of the second derivative with only a finite number of points? If not, can we determine a range of values the average second derivative must be in? What methods are there to approximate the average of the second derivative from a (finite) set of points?

If you can generalize your answer to nth derivatives, that would be even better.
You can't compute the exact average of the second derivative with just the function's values at the points.  You need the value of the derivatives at those points too.  Then the value is given by
$$\bar{f}''=\frac{f'(b)-f'(a)}{b-a}$$
The functions's value at a f finite set of points give you an average of the derivative, but no estimate of the value of the derivative at those points, so there is not enough information here.
You can't compute the exact average of the second derivative with just the function's values at the points.  You need the value of the derivatives at those points too.  Then the value is given by
$$\bar{f}''=\frac{f'(b)-f'(a)}{b-a}$$
The functions's value at a f finite set of points give you an average of the derivative, but no estimate of the value of the derivative at those points, so there is not enough information here.