Scaling is an important mathematical method in high school mathematics, especially in proving inequalities. With the decrease of the difficulty requirement of the series inequality in college entrance examination in recent years, the application focus of scaling method has gradually shifted from proving the series inequality to the derivative finale, especially in proving the derivative inequality. Here are some examples for your reference.

Zoom in and out by using basic inequality, and turn the curve into a straight line.

Use monotonicity to zoom in and out, and turn motion into stillness.

Using derivative to study monotonicity of function is an important method to prove elementary inequality.

The direct derivation of the proof method 1 proves that it is troublesome to judge the zero point of g( x) and find the minimum value f( x0) by finding f (x) because it contains the parameter m;

Prove 2 uses the monotonicity of logarithmic function y = ln x to make it static, which is simple and clear. In addition, this problem is also a classic example of dealing with the problem of hidden zeros of functions.

03

Flexible use of functional inequality to reduce complexity and simplify it.

There are two commonly used functional inequalities:

They originated from a set of exercises in high school textbooks (Elective 2-2, P32 in version A) and appeared in college entrance examination questions many times.