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 boundary point between the concave arc and the convex arc of a continuous 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.

The inflection point of the continuous curve y=f(x) on the interval I can be judged according to the following steps:

(1) Find f'' (x);

(2) Let f''(x)=0, solve the real root of this equation in interval I, and find out the point where f''(x) does not exist in interval I;

(3) For every point X found in [2] that has no real root or second derivative, check the adjacent symbols of f''(x) on the left and right sides of this point X. Then, when the symbols on both sides are opposite, this point (x, f(x)) is an inflection point, and when the symbols on both sides are the same, (x, f(x)) is not an inflection point.

Extended data:

Similar terms: stagnation correlation

For the image of two-dimensional function, the tangent plane of the stagnation point is parallel to the xy plane. It is worth noting that the stagnation point of a function is not necessarily the extreme point of this function (considering that the sign of the first derivative around this point is unchanged);

On the other hand, in a given area, the extreme points of a function are not necessarily the stagnation point (considering the boundary conditions), stagnation point (red) and inflection point (blue) of this function, and the stagnation point of this image is the local maximum or local minimum.