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If one considers the upper half circle as the graph of the function () =, then x = 0 is a critical point with critical value 1 due to the derivative being equal to 0, and x = ±1 are critical points with critical value 0 due to the derivative being undefined.
In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. One example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist.
A saddle point (in red) on the graph of z = x 2 − y 2 (hyperbolic paraboloid). In mathematics, a saddle point or minimax point [1] is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. [2]
This fold develops from a critical point defined by specific values of pressure, temperature, and molar volume. Because the surface is plotted using dimensionless variables that are formed by the ratio of each property to its respective critical value, the critical point is located at the coordinates (,,). When drawn using these dimensionless ...
In graph theory, a critical graph is an undirected graph all of whose proper subgraphs have smaller chromatic number. In such a graph, every vertex or edge is a critical element, in the sense that its deletion would decrease the number of colors needed in a graph coloring of the given graph. Each time a single edge or vertex (along with its ...
The points P 1, P 2, and P 3 (in blue) are collinear and belong to the graph of x 3 + 3 / 2 x 2 − 5 / 2 x + 5 / 4 . The points T 1, T 2, and T 3 (in red) are the intersections of the (dotted) tangent lines to the graph at these points with the graph itself. They are collinear too.
This illustrates the following rule: the topology of does not change except when passes the height of a critical point; at this point, a -cell is attached to , where is the index of the point. This does not address what happens when two critical points are at the same height, which can be resolved by a slight perturbation of f . {\displaystyle f.}
In network science, a critical point is a value of average degree, ... the network is complete (see complete graph). A random network is in a supercritical regime if ...