|
In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0 or undefined. For a differentiable function of several real variables, a critical point is a value in its domain where all partial derivatives are zero. The value of the function at a critical point is a critical value. The interest of this notion lies in the fact that the points where the function has local extrema are critical points. This definition extends to differentiable maps between R''m'' and R''n'', a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are also called ''bifurcation points''. In particular, if ''C'' is a plane curve, defined by an implicit equation ''f''(''x'',''y'') = 0, the critical points of the projection onto the ''x''-axis, parallel to the ''y''-axis are the points where the tangent to ''C'' are parallel to the ''y''-axis, that is the points where In other words, the critical points are those where the implicit function theorem does not apply. The notion of a ''critical point'' allows the mathematical description of an astronomical phenomenon that was unexplained before the time of Copernicus. A stationary point in the orbit of a planet is a point of the trajectory of the planet on the celestial sphere, where the motion of the planet seems to stop before restarting in the other direction. This occurs because of a critical point of the projection of the orbit into the ecliptic circle. ==Critical point of a single variable function== A critical point or stationary point of a differentiable function of a single real variable, ''f''(''x''), is a value ''x''0 in the domain of ''f'' where its derivative is 0: ''f'' ′(''x''0) = 0. A critical value is the image under ''f'' of a critical point. These concepts may be visualized through the graph of ''f'': at a critical point, the graph has a horizontal tangent and the derivative of the function is zero. Although it is easily visualized on the graph (which is a curve), the notion of critical point of a function must not be confused with the notion of critical point, in some direction, of a curve (see below for a detailed definition). If ''g''(''x'',''y'') is a differentiable function of two variables, then ''g''(''x'',''y'') = 0 is the implicit equation of a curve. A critical point of such a curve, for the projection parallel to the ''y''-axis (the map (''x'', ''y'') → ''x''), is a point of the curve where This means that the tangent of the curve is parallel to the ''y''-axis, and that, at this point, ''g'' does not define an implicit function from ''x'' to ''y'' (see implicit function theorem). If (''x''0, ''y''0) is such a critical point, then ''x''0 is the corresponding critical value. Such a critical point is also called a bifurcation point, as, generally, when ''x'' varies, there are two branches of the curve on a side of ''x''0 and zero on the other side. It follows from these definitions that the function ''f''(''x'') has a critical point ''x''0 with critical value ''y''0, if and only if (''x''0, ''y''0) is a critical point of its graph for the projection parallel to the ''x''-axis, with the same critical value ''y''0. For example, the critical points of the unit circle of equation ''x''2 + ''y''2 - 1 = 0 are (0, 1) and (0, -1) for the projection parallel to the ''y''-axis, and (1, 0) and (-1, 0) for the direction parallel to the ''x''-axis. If one considers the upper half circle as the graph of the function then ''x'' = 0 is the unique critical point, with critical value 1. The critical points of the circle for the projection parallel to the ''y''-axis correspond exactly to the points where the derivative of ''f'' is not defined. Some authors define the critical points of a function ''f'' as the ''x''-values for which the graph has a critical point for the projection parallel to either axis. In the above example of the upper half circle, the critical points for this enlarged definition are -1, 0 and -1. Such a definition appears, usually, only in elementary textbooks, when the critical points are defined before any definition of other curves than graphs of functions, and when functions of several variables are not considered (the enlarged definition does not extend to this case). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「critical point mathematics」の詳細全文を読む スポンサード リンク
|