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,

$\displaystyle R_{n,a}=\int^x_a\frac{f^{(n+1)}}{n!}(x-t)^ndt.$

The Lagrange form is,

$\displaystyle R_{n,a}=f^{(n+1)}(c)\frac{(x-a)^{(n+1)}}{(n+1)!}.$

If I create a Lagrange polynomial out of two points $\displaystyle (x_0,y_0)$ and $\displaystyle (x_1,y_1)$,
then according to my Numerical Analysis book the Lagrange form of the remainder is,

$\displaystyle f^{(n+1)}(c)\frac{(x-x_0)(x-x_1)}{(2)!}$,

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.