In calculus, stagnation point is also called stagnation point, stable point or critical point, which means that the first derivative of the function is zero, that is, at this point, the output value of the function stops increasing or decreasing. For the image of one-dimensional function, the tangent of the stagnation point is parallel to the X axis.

The stagnation point is the value of x obtained by taking the derivative of the original function and making it equal to 0. On the left and right sides of the stagnation point, the increment and decrement of the function change. If the stagnation point of the general quadratic function y = ax 2+bx+c is its vertex. At the stagnation point, the function can get the maximum, but it is not necessarily the maximum. The monotone interval of the function can be divided according to the stagnation point, that is, the monotonicity at the stagnation point may change.

The difference between extreme point, inflection point and stagnation point

1, extreme point: If f(a) is the maximum or minimum of function f(x), then a is the extreme point of function f(x), and the points of maximum and minimum are collectively called extreme points. The extreme point is the abscissa of the maximum or minimum point in the subinterval of the function image. The extremum point appears at the stagnation point (the point where the derivative is 0) or the non-derivative point of the function (the derivative function does not exist, so the extremum can be found, and the stagnation point does not exist at this time).

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

3. Inflection point: Also known as inflection point, it mathematically refers to the point that changes the upward or downward direction of the curve. 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).