Let f:[a,b]->R be a function that is not C^(n+1) on [a,b] but its n-th derivative is a Lipschitz function? How does the Lagrange's interpolating polynomial formula change? How does the error approximation change? How can i use it in order to find Simpson's and Newton's formulas of quadrature? I have searched a lot but I could not find a proper demonstration in any numerical analysis book. Please help!

Let $f:[a,b]\rightarrow R$ be a function that is not $C^{(n+1)}$ on $[a,b]$ but its $n$-th derivative is a Lipschitz function? How does the Lagrange's interpolating polynomial formula change? How does the error approximation change? How can I use it in order to find Simpson's and Newton's formulas of quadrature? I have searched a lot but I could not find a proper demonstration in any numerical analysis book. Please help!