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 $P_2$ - interpolation error estimate and the error bound of the Simpson rule in numerical integration.

翻译：本文推导了泰勒定理的一个变体，以得到一种新的极小化余项。对于定义在区间$[a,b]$上的给定函数$f$，该公式通过引入$f'$在$[a,b]$内$n+1$个等距点上的线性组合，并结合$f''(a)$和$f''(b)$来导出。随后，我们考虑了该泰勒类展开的两个经典应用：插值误差与数值求积公式。结果表明，采用该方法改进了拉格朗日$P_2$插值误差估计以及数值积分中辛普森法则的误差界。