好学高数（十三）：微分中值定理与导数的应用

　　01一、 微分中值定理及其应用1、微分中值定理的主要应用

　　(1)研究函数或导数的性态

　　(2)证明恒等式或不等式

　　(3)证明有关中值问题的结论

　　2、有关中值问题的解题方法

　　利用逆向思维,设辅助函数，一般解题方法:

　　(1)证明含一个中值的等式或根的存在，多用罗尔定理，可用原函数法找辅助函数。

　　(2)若结论中涉及含中值的两个不同函数,可考虑用柯西中值定理。

　　(3)若结论中含两个或两个以上的中值,必须多次应用中值定理。

　　(4)若已知条件中含高阶导数,多考虑用泰勒公式,有时也可考虑对导数用中值定理。

　　(5)若结论为不等式,要注意适当放大或缩小的技巧。

　　01 Differential mean value theorem and its application

　　1. Main application of differential mean value theorem

　　(1) Research the behavior of function or derivative (2) Prove identity or inequality

　　(3) Prove the conclusion of median problem

　　2. The solution method of median problem uses reverse thinking to set auxiliary functions. General solution method:

　　(1) Prove the existence of an equation or root containing a median. Use Rolle's theorem more often, and use the original function method to find auxiliary functions.

　　(2) If the conclusion involves two different functions with median value, Cauchy mean value theorem can be considered.

　　(3) If the conclusion contains two or more median values, the mean value theorem must be applied many times.

　　(4) If the known conditions contain high-order derivatives, consider using Taylor formula, and sometimes consider using the mean value theorem for derivatives.

　　(5) If the conclusion is inequality, pay attention to the technique of appropriately enlarging or shrinking.

　　02二、导数应用1、研究函数的性态:

　　增减，极值，凹凸，拐点，渐近线，曲率

　　2、解决最值问题

　　·目标函数的建立与简化

　　·最值的判别问题

　　3、其他应用：求不定式极限；几何应用;

　　相关变化率；证明不等式；研究方程实根等

　　4、补充定理

　　其他动漫

　　02 Derivative application

　　1. Study the properties of functions: increase and decrease, extreme value, concavity and convex, inflection point, asymptote, curvature

　　2. Solve the problem of maximum value

　　· Establishment and simplification of objective function

　　· Determination of maximum value

　　3. Other applications: Solve the limit of infinitives; Geometric applications; Relevant change rate; Prove inequality; Study the real roots of equations, etc.

　　4. Supplementary theorems