Функция туындысының нөлге теңелу нүктелері

Zero of the derivative of a function

In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name). For a differentiable function of several real variables, a stationary point is a point on the surface of the graph where all its partial derivatives are zero (equivalently, the gradient has zero norm). The notion of stationary points of a real valued function is generalized as critical points for complex valued functions. Stationary points are easy to visualize on the graph of a function of one variable: they correspond to the points on the graph where the tangent is horizontal (i. e., parallel to the x axis). For a function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the xy plane. The notion of a stationary point allows the mathematical description of an astronomical phenomenon that was unexplained before the time of Copernicus. A stationary point is the point in the apparent trajectory of the planet on the celestial sphere, where the motion of the planet seems to stop, before restarting in the other direction (see apparent retrograde motion). This occurs because of the projection of the planet orbit into the ecliptic circle.

A turning point of a differentiable function is a point at which the derivative has an isolated zero and changes sign at the point. A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). A turning point is thus a stationary point, but not all stationary points are turning points. If the function is twice differentiable, the isolated stationary points that are not turning points are horizontal inflection points. For example, the function has a stationary point at 1=x = 0, which is also an inflection point, but is not a turning point.

For the function f(x) = x4 we have (0) = 0 and (0) = 0. Even though (0) = 0, this point is not a point of inflection. The reason is that the sign of (x) changes from negative to positive. For the function f(x) = sin(x) we have (0) ≠ 0 and (0) = 0. But this is not a stationary point, rather it is a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign of (x) does not change; it stays positive. For the function f(x) = x3 we have (0) = 0 and (0) = 0. This is both a stationary point and a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign of (x) does not change; it stays positive. For the function f(x) = 0, one has (0) = 0 and (0) = 0. The point 0 is a non isolated stationary point which is not a turning point nor a horizontal point of inflection as the signs of (x) and (x) do not change. The function f(x) = x5 sin(1/x) for x ≠ 0, and f(0) = 0, gives an example where (x) and (x) are both continuous, (0) = 0 and (0) = 0, and yet f(x) does not have a local maximum, a local minimum, nor a point of inflection at 0. So, 0 is a stationary point that is not isolated.