1, the full name of derivative is derivative function. Because we like abbreviations so much, the value of derivative is also called derivative.
The result is very confusing, which confuses the concept of beginners.
Similar examples abound:
A, such as resistance, inductance, capacitance, reactance,,;
B, if uniform circular motion is uniform, uniform linear motion is conveying speed,,,,;
The culprit of these things is the teacher, who is lazy to clarify and likes abbreviations.
2. The geometric meaning of derivative function is to calculate the slope tangent, slope,
Gradient, the slope of the horizontal tangent is 0.
The slope with maximum or minimum value is 0, horizontal.
The slope of a straight line is also 0, so a slope of 0 is a necessary condition for having an extreme value or a maximum value.
3. It is not enough to infer whether the derivative is a maximum point or a minimum point. but we ... ...
Too many teachers often mislead students, especially in the sophomore year of multivariate function calculus.
Many professors still use necessary conditions to mislead students to discuss extreme points and calculate extreme values of multivariate functions.
For unary functions, we need to calculate the second derivative to be sufficient.
The combination of the two is the necessary and sufficient condition = the necessary and convenient condition.
This is what we usually call "if and only if", Iff = if and only if.
Final summary:
1, where the derivative is equal to 0, there will be a minimum and a maximum (this is for continuously differentiable function);
2. Where the derivative is equal to 0, there may be no maximum value or minimum value (such as a horizontal straight line parallel to the X axis).