To interpolate a function between points A, B, and C, where A and C provide function values and B provides first and second derivatives, a fourth degree polynomial can be used. This polynomial has five unknown coefficients, but only three equations can be derived from the known values and derivatives, making complete interpolation impossible. The remaining unknowns represent coordinates that could be anywhere, complicating the interpolation process. While traditional interpolation methods typically rely on function values, incorporating derivatives adds complexity and may require creative solutions. Ultimately, assumptions about the function's behavior across its domain are necessary to proceed with interpolation.
