The first derivative of the extreme point is 0, which describes the increase or decrease of the original function.
The second derivative at the inflection point is 0, which describes the concavity and convexity of the original function.
2. Different ways of interpretation.
If the function has first, second and third derivatives at this point and its domain, then the point where the first derivative of the function is 0 and the second derivative is not 0 is the extreme point; The point where the second derivative of the function is 0 and the third derivative is not 0 is the inflection point. For example, y = x 4 and x = 0 are extreme points rather than inflection points. If there is no derivative at this time, actual judgment is needed. For example, y = | x | x=0, the derivative does not exist, but x = 0 is the minimum point of the function.
Extended data:
If f(a) is the maximum or minimum value of the function f(x), then a is the extreme point of the function f(x), and the maximum and minimum points are collectively called extreme points. The extreme point is the abscissa of the maximum or minimum point in the subinterval of the function image. The extremum point appears at the stagnation point (the point where the derivative is 0) or the non-derivative point of the function (the derivative function does not exist, so the extremum can be found, and the stagnation point does not exist at this time).
Extreme point and stable point
The solution of the equation is called the stable point of the function.
Note: The derivative function is not required in the definition, so the extreme point of the derivative function must be a stable point, but the stable point is not necessarily an extreme point.
In mathematical analysis, the maximum and minimum values (maxima and minima) of a function are collectively called extremum (poles), and extremum is the maximum and minimum values (local or relative extremum) of a function within a given range or the global (global or absolute extremum) of a function. Pierre de Fermat was one of the first mathematicians to discover the maximum and minimum values of functions.
The inflection point, also known as the inflection point, refers to the point that changes the upward or downward direction of the curve in mathematics. Intuitively speaking, the inflection point is the point where the tangent intersects the curve (that is, the concave-convex boundary point of the curve). If the function of the graph has a second derivative at the inflection point, the second derivative has a different sign (from positive to negative or from negative to positive) or does not exist at the inflection point.
Let the function y=f(x) be continuous in a certain neighborhood of a point. If (,f ()) is the boundary point of the curve y=f(x), then (,f ()) is called the inflection point of the curve y=f(x).
Note: The inflection point (,f ()) is a point on the curve with abscissa and ordinate. Don't just take the abscissa as the inflection point.
References:
Baidu encyclopedia-limit point, Baidu encyclopedia-inflection point