The stagnation point, also known as stagnation point, stable point or critical point, is that the first derivative of the function is zero, that is, at this point, the output value of the function stops increasing or decreasing. 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).

The point where the first derivative of a function is 0 (stationary point is also called stable point, critical point). For multivariate functions, the stagnation point is the point where all the first-order partial derivatives are zero.

That is, at this point, the output value of the function stops increasing or decreasing. The stagnation point is not necessarily an extreme point, and the extreme point is not necessarily a stagnation point.

The stagnation point (red) and inflection point (blue) of this image are local maxima or local minima.

For the image of one-dimensional function, the tangent of the stagnation point is parallel to the X axis. For the image of two-dimensional function, the tangent plane of the stagnation point is parallel to the xy plane.

The stagnation point is not a point, similar to the extreme point, representing the x value of this point.

The difference between stagnation point and inflection point:

The term stagnation point of a function may be confused with the critical point of a given projection of a function graph.

The "critical point" is more general: the stagnation point of the function corresponds to the critical point of the projection parallel to the X axis. On the other hand, the key point of the projection parallel to the Y axis is the point where the derivative is not defined (more accurately, it tends to infinity). Therefore, some authors call the key points of these predictions "key points".

The inflection point is the point where the sign of the derivative changes. The inflection point can be a relative maximum or a relative minimum (also called local minimum and maximum). If the function is differentiable, then the inflection point is the fixed point; However, not all fixed points are inflection points. If the function is differentiable twice, then the fixed point of the fixed point is the horizontal inflection point.

For example, the function x3 has a fixed point at x=0, which is also an inflection point, but not a turning point.