Common deformations of sugar water inequalities include square inequalities, inverse inequalities, and trigonometric inequalities.

1. Square Inequality

Square inequality is the process of squaring the variables in a sugar water inequality, which can be done either by the square root property or by squaring both sides of the inequality simultaneously to get a new form of the inequality. For example, the inequality x > 2 is transformed into x^2 > 4. Sugar-water inequalities refer to a common class of inequalities in mathematics, and different forms of inequalities can be obtained by morphing them.

2. Inverse inequality

The inverse inequality is a sugar-water inequality in which the inequality symbols are flipped, i.e., the greater-than sign is changed to a less-than sign, the less-than sign is changed to a greater-than sign, and the inverse is taken on both sides of the inequality. For example, the inequality a > b is transformed into 1/a < 1/b.

3. Trigonometric inequality

Trigonometric inequality is to bring the variables in the sugar-water inequality into the trigonometric function, and utilize the trigonometric function to deform and solve the inequality. For example, for the sine function, we can use the monotonicity and periodicity of the sine function to work with the inequality. These deformation techniques are very useful in solving mathematical problems and proving mathematical theorems, and can be used flexibly according to specific problems and needs.

Sugar water inequality application:

1, inequality solution

Sugar water inequality is often used to solve inequality problems in mathematics, through the deformation of the inequality and simplification, you can find the solution set of inequality, that is, to satisfy the inequality conditions of the range of values of variables. By deforming a function with an inequality, important information such as the monotonicity of the function, zeros, extreme points, etc. can be obtained so that the image of the function can be plotted and its behavior can be studied. This is useful for solving math problems.

2. Maximum Value Problems

The sugar water inequality can be used to solve the maximum value problem, that is, to find the value of the variable that makes an expression reach its maximum or minimum value. By combining the objective function and the constraints into an inequality, you can use the method of sugar water inequality to solve the problem, so as to get the location and value of the most valuable point.

3, mathematical proof

Sugar water inequality also plays an important role in mathematical proof, through the use of sugar water inequality deformation method, you can derive some important mathematical inequalities and theorems, such as Cauchy-Schwartz inequality, mean value inequality, etc., which have a wide range of applications in the field of mathematical analysis, probability theory, geometry and other fields.