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：在本文中，我们推导了一种泰勒定理的变体，以获得新的最小化余项。对于定义在区间$[a,b]$上的函数$f$，通过在$[a,b]$中取$n+1$个等间隔点的$f'$的线性组合，以及$f''(a)$和$f''(b)$，得到了这个公式。然后，我们考虑了这种类似于泰勒展开的方法的两个应用：插值误差和数值积分公式。我们证明了使用这种方法可以改进Lagrange $P_2$ -插值误差估计和Simpson规则在数值积分中的误差界。