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A critical point of a function of a single real variable, f (x), is a value x 0 in the domain of f where f is not differentiable or its derivative is 0 (i.e. ′ =). [2] A critical value is the image under f of a critical point.
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.
Boca Raton, Florida, 2003; Section 6, Fluid Properties; Critical Constants. Also agrees with Celsius values from Section 4: Properties of the Elements and Inorganic Compounds, Melting, Boiling, Triple, and Critical Point Temperatures of the Elements Estimated accuracy for Tc and Pc is indicated by the number of digits.
Nevertheless, the word is still widely used, mainly for a concise formulation, for example: Critical locus, the set of the critical points of a differentiable function. Zero locus or vanishing locus, the set of points where a function vanishes, in that it takes the value zero. Singular locus, the set of the singular points of an algebraic variety.
At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point. (In fact, one can show that f takes both positive and negative values in small neighborhoods around (0, 0) and so this point is a saddle point of f.)
A line, usually vertical, represents an interval of the domain of the derivative.The critical points (i.e., roots of the derivative , points such that () =) are indicated, and the intervals between the critical points have their signs indicated with arrows: an interval over which the derivative is positive has an arrow pointing in the positive direction along the line (up or right), and an ...
The critical point is described by a conformal field theory. According to the renormalization group theory, the defining property of criticality is that the characteristic length scale of the structure of the physical system, also known as the correlation length ξ, becomes infinite. This can happen along critical lines in phase space.
If b 2 – 3ac = 0, then there is only one critical point, which is an inflection point. If b 2 – 3ac < 0, then there are no (real) critical points. In the two latter cases, that is, if b 2 – 3ac is nonpositive, the cubic function is strictly monotonic. See the figure for an example of the case Δ 0 > 0.