In advanced mathematics, a stagnation point refers to the point where the derivative of a function is zero , that is, the extreme point or inflection point of the function. When solving the maximum value, minimum value or inflection point of a function, it is necessary to find the stagnation point of the function.
Pay attention to the following points:
1. A zero derivative is not necessarily a stagnation point: Although a stagnation point is defined as a point where the derivative of a function is zero, this does not mean that all points with a zero derivative are stagnation points. Some points where the derivative is zero may be discontinuities or other special points of the function rather than stationary points.
2. The type of stagnation point: the stagnation point can be the maximum value point, minimum value point or inflection point of the function. By solving for the second derivative (or higher order derivatives), the type of stagnation point can be determined. When the second-order derivative is greater than zero, the stagnation point is a minimum point; when the second-order derivative is less than zero, the stagnation point is a maximum point; when the second-order derivative is equal to zero, it is necessary to further analyze the higher-order derivative to determine the stagnation point type .
Here are two examples:
Example 1: Consider the function f(x) = x^3 - 3x^2 + 2x. First derive f'(x) = 3x^2 - 6x + 2. Let f'(x) = 0 and solve for x = 1 or x = 2. These two points are the stationary points of the function. By solving the second order derivative f''(x) = 6x - 6, f''(1) = 0 can be obtained, indicating that x = 1 is an inflection point.
Example 2: Consider the function g(x) = x^4 - 4x^3 + 6x^2 - 4x + 1. Derivation gives g'(x) = 4x^3 - 12x^2 + 12x - 4. Let g'(x) = 0 and solve for x = 1. This point is the stationary point of the function. By solving the second derivative g''(x) = 12x^2 - 24x + 12, g''(1) = 0 can be obtained, indicating that x = 1 is an inflection point.