Generally [f (x1)+f (x2)]/2 >; The interval of f[(x 1+x2)/2] is called the concave interval of function f(x); On the contrary, it is a convex interval; The point where the concavity changes is called the inflection point.
General concavity is determined by the second derivative: f'' (x) >; The interval of 0 is a concave interval of f(x), otherwise it is a convex interval;
Example: find the convex-concave interval and inflection point of y = x 3-x 4.
Solution: y'=3x2-4x3, y'' = 6x-12x2;
Y''>0, get: 0
Therefore, the concave interval is (0,1/2); The convex interval is (-∞, 0), (1/2,+∞); The inflection point is (0,0), (1/2,116);
: Definition of function:
Given a number set A, assuming that the element in it is X, now apply the corresponding rule F to the element X in A, and record it as f(x), and get another number set B. Assuming that the element in B is Y, the equivalent relationship between Y and X can be expressed as y=f(x). We call this relationship a functional relationship, or function for short. The concept of function includes three elements: definition field A, value field C and corresponding rule F, among which the core is corresponding rule F, which is the essential feature of function relationship. ?
Function was originally translated by China mathematician Li in his book Algebra in Qing Dynasty. He translated this way because "whoever believes in this variable is the function of that variable", that is, the function means that one quantity changes with another quantity, or that one quantity contains another quantity.
The definition of function is usually divided into traditional definition and modern definition. The essence of these two functional definitions is the same, but the starting point of narrative concept is different. The traditional definition is from the perspective of movement change, and the modern definition is from the perspective of set and mapping.