nLab real cubic function

Definition

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

$$\frac{d^4 f}{d x^4} = 0$$

with initial conditions

$$f(0) = d$$

$$\frac{d f}{d x}(0) = c$$

$$\frac{d^2 f}{d x^2}(0) = 2 b$$

$$\frac{d^3 f}{d x^3}(0) = 6 a$$

Properties

Extrema

The derivative of a real cubic function $f:\mathbb{R} \to \mathbb{R}$, $f(x) \coloneqq a x^3 + b x^2 + c x + d$ for real numbers $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$, $d \in \mathbb{R}$ is the function $\delta_x f:\mathbb{R} \to \mathbb{R}$, $\delta_x f(x) \coloneqq 3a x^2 + 2b x + c$. The discriminant of the derivative is given by $\Delta_\partial = (2b)^2 - 4 (3a) c = 4b^2 - 12a c$. If the discriminant of the derivative is positive $\Delta_\partial \gt 0$, then there are two local extrema in the cubic function. Otherwise, $f$ is monotonic on the entire domain of $a \gt 0$, or antitonic on the entire domain if $a \lt 0$.

Using the smooth quadratic formula?, we find that for positive discriminant the two extrema are located at

$$x_1 = \frac{-2b + \mathrm{sqrt}_\mathrm{sm}(4b^2 - 12a c)}{6a} = \frac{-2b + 2 \mathrm{sqrt}_\mathrm{sm}(b^2 - 3a c)}{6a} = \frac{-b + \mathrm{sqrt}_\mathrm{sm}(b^2 - 3a c)}{3a}$$

$$x_2 = \frac{-2b - \mathrm{sqrt}_\mathrm{sm}(4b^2 - 12a c)}{6a} = \frac{-2b - 2 \mathrm{sqrt}_\mathrm{sm}(b^2 - 3a c)}{6a} = \frac{-b - \mathrm{sqrt}_\mathrm{sm}(b^2 - 3a c)}{3a}$$

For negative discriminant, there is no extrema. For zero discriminant, there is a saddle point? at

$$x = -\frac{b}{3a}$$

If $a = 0$, the real cubic function is degenerate?; it becomes a real quadratic function.

Inflection point

THe inflection point of the real cubic function $f:\mathbb{R} \to \mathbb{R}$, $f(x) \coloneqq a x^3 + b x^2 + c x + d$ for real numbers $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$, $d \in \mathbb{R}$ occurs at the zero of the second derivative of $f$:

$$\partial_x f(x) = 6 a x + 2b = 0$$

which occurs at

$$x = -\frac{b}{3a}$$

The inflection point is to real cubic functions what the extremum was to real quadratic functions.

In constructive mathematics

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 $f:\mathbb{R} \to \mathbb{R}$ such that one cannot decide whether $f$ has two local extrema, a saddle point, or zero local extrema. Similarly, there exists real cubic functions $f:\mathbb{R} \to \mathbb{R}$ such that one cannot decide whether the inflection point occurs to the left, the right, or on the $y$-intercept line.

Depressed cubic functions

An unnormalized depressed cubic function is a real cubic function whose inflection point occurs at $x = 0$, and has the canonical form of $g(x) \coloneqq a x^3 + p x + q$, with $\vert a \vert \gt 0$. $g$ is called a depressed cubic function when $a$ is normalized to $1$.

Every real cubic function $f:\mathbb{R} \to \mathbb{R}$, $f(x) \coloneqq a x^3 + b x^2 + c x + d$ for real numbers $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$, $d \in \mathbb{R}$, is a translation of an unnormalized depressed cubic function $g:\mathbb{R} \to \mathbb{R}$ by the equation

$$g(x) = f\left(x - \frac{b}{3a}\right)$$

$$a x^3 + p x + q = a \left(x - \frac{b}{3a}\right)^3 + b \left(x - \frac{b}{3a}\right)^2 + c \left(x - \frac{b}{3a}\right) + d$$

$$a x^3 + p x + q = a x^3 - 3 a \frac{b}{3a} x^2 + 3 a \left(\frac{b}{3a}\right)^2 x - a \left(\frac{b}{3a}\right)^3 + b x^2 - 2 b \frac{b}{3a} x + b \left(\frac{b}{3a}\right)^2 + c x - c \left(\frac{b}{3a}\right) + d$$

$$a x^3 + p x + q = a x^3 + \frac{b^2}{3a} x - \frac{b^3}{27a^2} - \frac{2 b^2}{3a} x + \frac{b^3}{9 a^2} + c x - \frac{b c}{3a} + d$$

$$a x^3 + p x + q = a x^3 + \frac{3a c - b^2}{3a} x + \frac{2 b^3 - 9 a b c + 27 a^2 d}{27a^2}$$

thus,

$$p = \frac{3a c - b^2}{3a}$$

$$q = \frac{2 b^3 - 9 a b c + 27 a^2 d}{27a^2}$$

Discriminant

The discriminant of an unnormalized depressed cubic function $g(x) \coloneqq a x^3 + p x + q$ is given by

$$\Delta = -4 p^3 - 27 a q^2$$

Substituting in the values above for $p$ and $q$, the discriminant of a general real cubic function $f:\mathbb{R} \to \mathbb{R}$, $f(x) \coloneqq a x^3 + b x^2 + c x + d$ for real numbers $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$, $d \in \mathbb{R}$ is

$$\Delta = -4 \left(\frac{3a c - b^2}{3a}\right)^3 - 27 a \left(\frac{2 b^3 - 9 a b c + 27 a^2 d}{27a^2}\right)^2 = \frac{4(b^2 - 3a c)^3 - (2 b^3 - 9 a b c + 27 a^2 d)^2}{27 a^3}$$

Exact zeroes

Partial inverse functions

Given a real cubic function $f(x) \coloneqq a x^3 + b x^2 + c x + d$ for real numbers $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$, $d \in \mathbb{R}$ such that $\vert a \vert \gt 0$, the following cases are possible:

• If the discriminant of the derivative of $f$ is positive, $\Delta_\partial \gt 0$, then there are three branches of the partial inverse function of $f$.

• If the discriminant of the derivative of $f$ is zero, $\Delta_\partial = 0$, then there are two branches of the partial inverse function of $f$.

• If the discriminant of the derivative of $f$ is negative, $\Delta_\partial \lt 0$, then there is only one branch of the partial inverse function of $f$.

In all cases, the partial inverse functions are solutions of the first-order nonlinear ordinary differential equation

$$(3 a F^2 + 2 b F + c) \frac{d F}{d x} = 1$$

with specific initial conditions.

See also

• real number

• cubic function

• real quadratic function

• real polynomial function

References

See also:

• Wikipedia, Cubic function
• Wikipedia, Cubic equation