高等数学微分中值定理的应用泰勒公式

第一步：证明a附近存在一点N，使得导数大于0，且f(N)>0；
第二步：证明（N,b）之间存在一点M，使得导数小于0；原因是：f在[N,b]上从正数 f(N)过渡到 f(b)=0, 从而由中值定理，(N,b)内必定有一点使得导数小于0;
第三步：因此[N,M]上面，导函数从正过渡到负，根据中值定理，必定存在一点，使得二阶导数小于0；

二阶可导，说明一阶导数在（a，b）连续，
由罗尔中值定理，存在 c∈（a，b）使 f '(c) = 0，
f '(x) 在 [a，c] 上连续，在（a，c）上可导，由拉格朗日中值定理，
存在 ξ∈(a，c) 使 f ''(ξ) = [f '(c) - f '+(a)] / (c-a) < 0 。

On the Mean Value Theorem and Taylor's formula
Mean Value Theorem and Taylor's formula is the basic formula of differential calculus, which constitute an important part of the basic theory of calculus. Mean Value Theorem is a advantaged(powerful) tool to research functions' own nature(properties) on the interval which take advantage of the properties of functions. It includes: Rolle theoreom; Lagrange mean value theorem; Cauchy Mean Value Theorem. Taylor's formula is an important in mathematical analysis, which is widely used in the calculation and proof of a number of important issues(problems). This article describes some of their applications.
Mean Value Theorem; Taylor formula; limits; inequalities.
单复数可以调整下，细节可以调整下，句型变化还有许多的。