A new second order Taylor-like theorem with an optimized reduced remainder

In this paper, we derive a variant of the Taylor theorem to obtain a new minimized remainder. For a given function f defined on the interval [a,b], this formula is derived by introducing a linear combination of f^′ computed at n+1 equally spaced points in [a,b], together with f^′′(a) and f^′′(b). We then consider two classical applications of this Taylor-like expansion: the interpolation error and the numerical quadrature formula. We show that using this approach improves both the Lagrange P₂- interpolation error estimate and the error bound of the Simpson rule in numerical integration.