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种变体，以获得一种新的最小化余项。对于定义在区间$[a,b]$上的给定函数$f$，通过引入$f$在$[a,b]$中$n+1$个等距点处计算的$f'$的线性组合以及$f''(a)$和$f''(b)$，推导出该公式。随后我们考虑这一泰勒型展开的两种经典应用：插值误差和数值求积公式。研究表明，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。