If there is always f ((a+b)/2) >: (f(a)+f(b))/2, then the graph of f(x) on d is convex (or convex arc).
Steps for finding convexity and inflection point
(1) Domain found;
(2) Find the second derivative of f(x) (in the form of product);
(3) Find the point where the second derivative of f(x) is equal to 0 and the point where the second derivative of f(x) does not exist;
(4) Divide the domain into several cells with the above points, and judge its concavity by looking at the sign of the second derivative of f(x) between each cell (greater than zero is a concave function, less than zero is a convex function);
(5) If the sign of the second derivative of f(x) is different on both sides of point X, then (x, f(x)) is an inflection point, otherwise it is not (that is, the first sufficient condition of the inflection point mentioned in the derivative).
Extended data
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 intuitively understand the meaning of "concave" and "convex", so we can only use expressions. Of course, n-dimensional expression is more complicated than two-dimensional affirmation.
However, it is the same objective fact whether it is intuitively understood from graphics or from expression. 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