Polynomial with equidistant roots
The Lagrange interpolation error involves a polynomial L(x) with roots at the nodes used for interpolation. For the case of equidistant nodes, the error is large near the end points. This situation can be improved by choosing non-equidistant nodes, the case of Chebyshev nodes being a quasi-optimal choice.
Shift any of the nodes A, B, C, D to obtain a reduction in the peaks observed near the end points.