Basic inequality is an important mathematical formula, which is widely used in solving and proving inequalities. The formula shows that for any nonnegative real numbers A and B, there is (a+b)? ≥4ab .

The formula can also be written as:

Answer? +2ab+b？ ≥4ab or: a? -2ab+b？ ≥0

This formula can be derived from the complete square formula, which points out the relationship between the square of the sum of two non-negative real numbers and its product. If and only if a=b, the equal sign holds.

This formula has many applications, such as proving that the average value of two numbers is not less than their geometric average value, and proving that the length of the bisector of the inner angle of a triangle is not less than half of the circumference. Mastering basic inequalities is the basis of mathematics learning, and it is also necessary for advanced mathematics knowledge.

Basic inequality occupies a very important position in mathematics, which is an important part of inequality theory and one of the classical theorems in mathematics. In fact, basic inequality is an important inference and tool of a large number of inequalities, so we can use this formula to solve many mathematical problems.

One of the most common applications of basic inequalities is to verify inequalities. Many mathematical inequalities can be proved by using basic inequalities, which can sometimes simplify the proof process and make the proof more concise and easy to understand. Especially in mathematical competitions, basic inequality is an essential tool to solve inequality problems, and sometimes a basic inequality can solve multiple inequality problems.

In addition, basic inequalities can also be applied to optimization problems. In the optimization problem, the minimum or maximum value of a function is required, and the basic inequality just provides a suitable method. For example, we can prove the maximum or minimum of an optimization problem with some constraints by applying basic inequalities.

In a word, basic inequality is an important tool in mathematics, which not only plays a key role in the proof of inequality problems, but also has high practical value in optimization problems. Mastering the application of basic inequalities and skillfully using them is of great significance for improving our mathematical literacy and solving various problems.