判定函数

At the beginning of this thesis, the author gives the definition and the equivalent definition of convex function, and then proves the equivalent relationship between them. Secondly the author proposes the decision theorem of convex function which provides a judgment basis of whether a function is a convex function. Thirdly the author summarizes and proves the convex function's operational, basic, differential and integral property. Finally the author proves several famous convex function inequalities, such as Jensen inequality, Holder inequality, Cauchy inequality. The author also provides the application of these inequalities and illustrates the importance of convex function's basic inequality and integral property in the proving process.

本文开始给出了凸函数的定义及等价定义，并证明了它们之间的等价关系；接着提出了凸函数的判定定理，对一个函数是否是凸函数提供判断依据；然后对凸函数的运算性质、基本性质、微分性质、积分性质四个方面的性质进行了总结，并给予了证明；最后证明了凸函数的几个著名不等式詹森不等式、赫尔德不等式、柯西不等式，给出了这几个不等式的一些应用实例，并举例说明凸函数的基本性质和积分性质在不等式证明过程中的重要作用。

At the beginning of this thesis, the author gives the definition and the equivalent definition of convex function, and then proves the equivalent relationship between them. Secondly the author proposes the decision theorem of convex function which provides a judgment basis of whether a function is a convex function. Thirdly the author summarizes and proves the convex function's operational ,basic , differential and integral property. Finally the author proves several famous convex function inequalities, such as Jensen inequality, Holder inequality, Cauchy inequality and Minkowski inequality. The author also provides the application of these inequalities and illustrates the importance of convex function's basic inequality and integral property in the proving process.

本文开始给出了凸函数的定义及等价定义，并证明了它们之间的等价关系；接着提出了凸函数的判定定理，对一个函数是否是凸函数提供判断依据；然后对凸函数的运算性质、基本性质、微分性质、积分性质四个方面的性质进行了总结，并给予了证明；最后证明了凸函数的几个著名不等式詹森不等式、赫尔德不等式、柯西不等式和闵可夫斯基不等式以及这几个不等式的应用，并举例说明凸函数的基本性质和积分性质在不等式证明过程中的重要作用。

This article according to the function image axial symmetry and the point symmetry basic principle,the use function taylor series expansion,determines and seeks the polynomial function image the axis of symmetry or the center of symmetry.

根据函数图象轴对称及点对称的基本原理，利用函数的泰勒级数展开式，来判定和寻找函数图象的对称轴或对称中心。

According to the some properties of the differentiation ,I found that the symmetry of the graph is closely related to the first-order differentiation and the second-order differentiation .from this ,I search for a way to identify the symmetry of the functional graph for the direct line or the central symmetry of the functional graph for the some dot .

摘 要：利用导数的一些性质，发现了函数图象的对称性与函数的一阶、二阶导数的密切关系。根据这些关系，找到了一种判定函数图象是否关于某一直线对称或关于某点成中心对称的方法。

This paper reveals the relationship between the parity of derivative function and the symmetry of the graph function, finding a effective way to judge whether the functional graph can have the axial symmetry or central symmetry.

揭示了函数导数的奇偶性与函数图像对称性的关系，发现了一种可以用来判定函数图像是否具有轴对称性或中心对称性的有效方法。

Methods According to chemical equilibrum principle and the use of derivative in judging the increasing and decreasing of function. this influence has been discussed.

依据化学平衡原理和导数在判定函数增减性方面的应用，分析了反应物初始浓度变化对转化率的影响。

With the discussion of extreme value problem of the multiple-valued function,and the popularizing of extreme value theory of the two-value function,This paper obtains differentiation rule about the extreme value of the multiple-value function,and illustrates the apllication of the theory with the example,providing proper guidance for the teaching of calculus.

文章讨论了多元函数的极值问题，并将二元函数的极值判定定理加以推广，得到关于多元函数极值的判定法则，并举例说明其应用，这对微积分的教学有一定的指导意义。

And it makes a detailed and brief exposition, and offers some extremely targeted examples of the application, in order to understand the integrability conditions and enhance the understanding and application capability.

文章最后专门讨论了复合函数的黎曼可积性和可积函数列的逐项积分，得出了如何根据特定条件来判断一个复合函数可积性的定理和判定一个函数列可逐项积分的一个充分条件，并将其推广，得到一个更弱的充分条件。

In this paper a decision method of uniform convergence for function sequence is given, and a few of properties for the uniformly convergent function-sequence are obtained by this method, which is used in the construction of a new uniformly convergent function-sequence by others known.

给出了一个定理用于判定函数列的一致收敛，通过此判定定理，又得出了由一致收敛函数列构造新的一致收敛函数列的几个性质。

Some methods of deciding the extreme values for the monadic functions and two-place functions were applied to decide the extreme values for the multivariate functions and then some effective methods of deciding the extreme values for the multivariate functions were presented.

将一元函数和二元函数极值的部分判别方法推广到多元函数极值的判别，提出了判定多元函数极值的几个方法。

What are your goals and strategies for growth?

你的成长目标和策略是什么？

And unto the angel of the church in Sardis write; These things saith he that hath the seven irits of God, and the seven star I know thy works, that thou hast a name that thou livest, and art dead.

3：1 你要写信给撒狄教会的使者，说，那有神的七灵和七星的，说，我知道你的行为，按名你是活的，其实是死的。