What are the methods of proving inequalities, and how to prove basic inequalities

Comparative, synthetic, analytical, inductive, deflationary

Here's how to prove the fundamental inequality:

1. Comparative method: including two methods: difference and quotient.

2. Comprehensive law.

When proving inequality, starting from the known conditions of the proposition, using axioms, theorems, laws, etc., to gradually deduce the proposition to be proved is called the synthesis method, which is a method of deriving cause and effect.

3. Analytical method.

When proving the inequality, starting from the proposition to be proven, analyzing the sufficient conditions to make it true, using some known basic principles, gradually exploring, and finally reducing the conditions for the proposition to be true to a theorem, simple fact or the condition of the proposition, this method of proof is called the analytical method, which is the method of holding the cause of effect.

4. Deflation method.

When proving an inequality, sometimes the value of the inequality to be proved is appropriately enlarged or reduced according to the need, so that it can be simplified and difficult to achieve the purpose of proof, which is called the scaling method.

5. Mathematical induction.

To prove inequality by mathematical induction, it is necessary to pay attention to two steps and one conclusion.

In the second step of proof, comparison, deflation and analysis are generally used.

6. Counter-evidence.

When proving inequality, first assume that the opposite of the proposition to be proved is true, combine it as a condition with other conditions, and use the basic principles such as known definitions, theorems, and axioms to gradually deduce a conclusion that contradicts the conditions of the proposition or the proven theorem or the recognized simple facts, so as to show that the conclusion of the null hypothesis is not true, and thus affirm the validity of the conclusion of the original proposition is called the method of counterproof.

Fundamental inequality.

There are 20 ways to prove it.

The main ones are: 1. Proof of difference.

The difference proof is for a unary one-time inequality.

Build a unary function. When encountering the inequality problem, we should first observe the type of inequality in combination with the properties of the inequality, and after determining that it is a unary primary inequality problem, we can construct a unary function to solve it by the difference method.

2. Analytical proof.

Analytical proof is also called "reverse deduction method" or "causal cause method". It starts from the conclusion to be proved, analyzes the conditions that make it true, that is, seeks the sufficient conditions to make each step true.

In the end, the conclusion to be proved is reduced to the conditions (known conditions, theorems, definitions, axioms, etc.) that determine the obvious validity of a stocking manuscript.

3. Comprehensive law certification.

Comprehensive proof is a logical reasoning method that deduces from the known to the unknown, and from the known conditions to the conclusion.

1. Trigonometric inequality.

Trigonometric inequalities, i.e. in which the sum of the two sides of a triangle is greater than the third side, sometimes also refers to formulas containing trigonometric functions connected by inequality signs (not described here). Although trigonometric inequalities are simple, they are the most fundamental conclusions of plane geometric inequalities.

2. Mean inequality.

The mean inequality, also known as the mean inequality, is an important formula in mathematics. The content of the formula is hn gn an qn, that is, the harmonic mean does not exceed the geometric mean, the geometric mean does not exceed the arithmetic mean, and the arithmetic mean does not exceed the square mean.

3. Cauchy inequality.

The Cauchy inequality was obtained by the great mathematician Cauchy when he was working on the problem of "flow numbers" in mathematical analysis.

Historically, however, this inequality should be called the Cauchy-Buniakowsky-Schwarz inequality, because it was the latter two mathematicians, independently of each other, who generalized it in integralism, that led to the near-perfection of the inequality.

Cauchy inequality is an inequality discovered by Cauchy in the process of research, which has a very wide application in solving the related problems of inequality proof, so it is very important in the improvement of higher mathematics and is one of the research contents of higher mathematics.

4. Geometric mean inequality.

The root number ab, called the geometric mean, embodies a geometric relationship, that is, through any point on the diameter of a circle to make a perpendicular line, the diameter is separated by two parts of a, b, then half of the length of the perpendicular line in the circle is the root number ab, and (a+b) 2 root number ab!This is what it means geometrically, and why it is called a geometric mean.

The arithmetic-geometric mean inequality, abbreviated as arithmetic inequality, is a common and fundamental inequality that expresses a constant inequality between the arithmetic mean and the geometric mean.

5. Young's inequality.

Young's inequality is also known as Young's inequality, which is a special case of weighted arithmetic mean inequalities, and Young's inequality is a quick way to prove Holder's inequality.

Methods of proving inequalities: there are comparative methods, synthesis methods, analytical methods, deflation methods, mathematical induction methods, counterproof methods, commutation methods, construction methods, etc.

1. Difference comparison method: according to a-b>0 a>b, if you want to prove a>b, you only need to prove a-b>0.

2. Element exchange method: The purpose of element exchange is to reduce the number of variables in the inequality, so as to make the problem difficult and simple.

3. Comparison of business quotients: according to a b=1, when b > 0, a>b is obtained; When b>0, to prove a>b, you only need to prove ab>1;While.

b 0, a b.

4. Comprehensive method: cause leads to effect. When proving an inequality, the inequality to be proved is deduced by using the properties of the inequality and the appropriate deformation from the known inequality and the conditions of the problem. Legitimacy is also known as the method of inference or the method of cause and effect.

The proof methods include comparative method, synthesis method, analysis method, deflation method, mathematical induction method, counterproof method, commutation method, construction method, etc.

Difference comparison method: according to a-b>0 a>b, to prove a>b, only a-b>0 is required. Swap Method: The purpose of swapping is that.

Reduce the number of inequalities to make the problem easier and more complex. Inequality proof is a very important content, in the quantitative relationship, in the analysis of inequality proof problems, looking for ways to solve (proof) problems, advocate the use of comprehensive method and analysis method at the same time, just like digging a cave, from the two ends to the middle, so as to shorten the distance between the condition and the conclusion.

Inequality proof method:

Comparative method: Differential comparison method: according to a-b>0 a>b, if you want to prove a>b, you only need to prove a-b>0;Comparative Approach to Trading:

According to a b=1, when b > 0, a>b is obtained; When b>0, to prove a>b, you only need to prove ab>1;When b<0, we get a<>

The synthesis method is a method of proving that leads to cause and effect. When using the comprehensive method to prove the inequality, we should pay attention to observing the structural characteristics of the inequality and choose the appropriate formula as the basis, among which the mean inequality is the most commonly used, the proof method uses the ternary mean inequality to prove once or twice, and the second method mainly uses the property proof of the inequality.