# Derivative error for Lagrange interpolation

I was reading a book and I found this (with some context):

if $f(x)=L(x)+R(x)$, with $L$ the quadratic interpolation with three points $x_0, x_1$ and $x_2$, then $R(x)=\dfrac{f'''(\epsilon(x))}{6} (x-x_0)(x-x_1)(x-x_2)$, for some $x_0<\epsilon(x)<x_2$, and $R'(x)=\dfrac{f^{(4)}(\mu)}{24} (x-x_0)(x-x_1)(x-x_2)+\dfrac{f'''(\epsilon(x))}{6}D((x-x_0)(x-x_1)(x-x_2))$ where D is the derivative operator and $\mu$ was not described.

How do I prove it?

• My question is about the derivative of R and not about the R. – Renan Willian Prado Apr 8 '16 at 5:12
• Search on Google "line search.pdf " and it's the first PDF, then go to the page 95. This is where i read it. This is a part of the book "Optimization theory and methods: nonlinear programming." – Renan Willian Prado Apr 8 '16 at 18:08