I derive the Lagrange form of the remainder by using the Weighted Mean-Value theorem on the integral form of the remainder.
The integral form according to my Calculus book is,
The Lagrange form is,
If I create a Lagrange polynomial out of two points and ,
then according to my Numerical Analysis book the Lagrange form of the remainder is,
which to me makes more sense since we should take into account all points when calculating the remainder.
Please let me know if this question does not make sense and I will try and rephrase.