[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"