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How to judge the maximum value and minimum value with the derivative of needle and thread
According to the derivative of the function, the specific methods for judging the maximum and minimum are as follows:

[1] Let the function y=f[x], then y'=f'[x].

[2] When making an image of y'=f'[x], every intersection with the X axis is the extreme point of y=f[x].

[3] As for the maximum or minimum, it can be judged according to whether the "needle threading" is bottom-up (minimum) or top-down (maximum).

Refer to Figure 1:

In addition, the "thread method" is also applied in junior high school mathematics, such as:

Add x 3-2x 2-x+2 >: 0 to (x-2)(x- 1)(x+ 1) >; 0

The roots of (x-2)(x- 1)(x+ 1)=0 are: x 1 = 2, x2 = 1, x3 =- 1.

Refer to Figure 2:

Note: Needle-piercing method, also known as "multi-axis root-piercing method" or "using several axes method"