symmetric monoidal (∞,1)-category of spectra
analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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A real cubic function is a cubic function in the real numbers. Equivalently, it is a solution to the fourth-order linear homogeneous ordinary differential equation
with initial conditions
The derivative of a real cubic function , for real numbers , , , is the function , . The discriminant of the derivative is given by . If the discriminant of the derivative is positive , then there are two local extrema in the cubic function. Otherwise, is monotonic on the entire domain of , or antitonic on the entire domain if .
Using the smooth quadratic formula?, we find that for positive discriminant the two extrema are located at
For negative discriminant, there is no extrema. For zero discriminant, there is a saddle point? at
If , the real cubic function is degenerate?; it becomes a real quadratic function.
THe inflection point of the real cubic function , for real numbers , , , occurs at the zero of the second derivative of :
which occurs at
The inflection point is to real cubic functions what the extremum was to real quadratic functions.
In classical mathematics, the law of trichotomy holds in the real numbers, so the three cases above cover every real number. However, in constructive mathematics, trichotomy does not hold in the real numbers, and as a result, there exists real cubic functions such that one cannot decide whether has two local extrema, a saddle point, or zero local extrema. Similarly, there exists real cubic functions such that one cannot decide whether the inflection point occurs to the left, the right, or on the -intercept line.
An unnormalized depressed cubic function is a real cubic function whose inflection point occurs at , and has the canonical form of , with . is called a depressed cubic function when is normalized to .
Every real cubic function , for real numbers , , , , is a translation of an unnormalized depressed cubic function by the equation
thus,
The discriminant of an unnormalized depressed cubic function is given by
Substituting in the values above for and , the discriminant of a general real cubic function , for real numbers , , , is
Given a real cubic function for real numbers , , , such that , the following cases are possible:
If the discriminant of the derivative of is positive, , then there are three branches of the partial inverse function of .
If the discriminant of the derivative of is zero, , then there are two branches of the partial inverse function of .
If the discriminant of the derivative of is negative, , then there is only one branch of the partial inverse function of .
In all cases, the partial inverse functions are solutions of the first-order nonlinear ordinary differential equation
with specific initial conditions.
See also:
Last revised on June 2, 2022 at 11:33:45. See the history of this page for a list of all contributions to it.