Let the function f(x) be defined on the interval I, if any two points x in I? And what about x? , and any λ∈(0, 1), all have:

f(λx？ +( 1-λ)x？ )& gt=λf(x？ )+( 1-λ)f(x？ ),

Then f is called a convex function on I, and if the inequality is strictly true, it is called ">". If not, it is called f(x) as a strictly convex function on I..

Similarly, if "> =" is replaced by "

2. From the geometric point of view, it is:

Take any two points on the image of function f(x). If the part of the image between these two points is always below the line segment connecting these two points, then this function is a concave function. Similarly, if the part between these two points is always above the line segment connecting these two points, then the function is convex.

Intuitively, the convex function is that the image protrudes upward.

If the function f(x) is second-order differentiable in the interval I, the necessary and sufficient condition for f(x) to be convex in the interval I is f "(x); =0。

Extended data:

Different statements:

But by the way, the definition of concavity and convexity of function in China's mathematics circle is contrary to many definitions abroad. Concave and convex in domestic textbooks refer to curves, not functions. The concavity and convexity of the image is consistent with the intuitive feeling, but contrary to the concavity and convexity of the function. As long as you remember that "the concavity and convexity of a function is opposite to the concavity and convexity of a curve", you will not confuse the concept.

In addition, the views on concavity and convexity in textbooks and guidance books of different disciplines in China are also opposite. Generally speaking, it can be accurately interpreted as:

1、f(λx 1+( 1-λ)x2)& lt； = λ f (x1)+(1-λ) f (x2), that is, V-shape, that is, "convex to the origin" or "convex downward";

2, f (λ x1+(1-λ) x2) > = λ f (x1)+(1-λ) f (x2), that is, type A, that is, "concave origin" or "convex origin".

It is clear that the convex/concave points to the origin. There is no ambiguity in the statement that the upper part is convex and the lower part is concave.

In the two-dimensional environment, that is, in the plane rectangular coordinate system, we can intuitively see whether a two-dimensional curve is convex or concave by drawing. Of course, it also corresponds to an analytical expression, which is inequality.

In the case of multi-dimensions, graphics can't be drawn, so we can't understand the meaning of "concave" and "convex" intuitively, only through expressions. Of course, the expression of n-dimension is more complicated than the affirmation of two-dimension, but whether we understand it intuitively from graphics or expressions, we are all describing the same objective fact.

Moreover, the concavo-convex defined according to the function diagram is just the opposite of the concavo-convex defined according to the function.

Baidu encyclopedia-concavity and convexity of function