Cubic function

Graph of a cubic function; the zeros (y = 0) are where the graph intersects the x -axis . The graph has two extreme points .

Graph of the cubic function f (x) = 1-x + x² + x³

The three roots of the cubic function f (x) = 1-x + x² + x³ in the Gaussian plane

In mathematics , a cubic function is understood to be a completely rational function of the 3rd degree, i.e. a function on real numbers that is in the form ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$

${\displaystyle f(x)=ax^{3}+bx^{2}+cx+d}$

can be written with and . ${\displaystyle a,b,c,d\in \mathbb {R} }$ ${\displaystyle a\neq 0}$

Cubic functions as a real polynomial functions of polynomials over to be construed. ${\displaystyle \mathbb {R} }$

properties

Behavior in infinity

As with all completely rational functions of odd degree applies

${\displaystyle \lim \limits _{x\to +\infty }f(x)=+\infty }$, , ${\displaystyle \lim \limits _{x\to -\infty }f(x)=-\infty }$

if the leading coefficient is positive, and ${\displaystyle a}$

${\displaystyle \lim \limits _{x\to +\infty }f(x)=-\infty }$, , ${\displaystyle \lim \limits _{x\to -\infty }f(x)=+\infty }$

if is negative. ${\displaystyle a}$

zeropoint

Since a cubic function is continuous as a polynomial function , it follows from the behavior at infinity and the intermediate value theorem that it always has at least one real zero. On the other hand, a completely rational function of degree cannot have more than zeros. Thus it follows: A cubic function has at least one and a maximum of three zeros . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} }$

To find the zeros of a cubic function, see Cubic Equation and Cardan Formulas . The discriminant of the general cubic function is ${\displaystyle f}$

${\displaystyle D=b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd}$

and is suitable for the classification of zeros of the polynomial : In the case there are three different real zeros, in the case only one. It is true that there is either a single and a double real zero or there is a triple real zero. ${\displaystyle D>0}$ ${\displaystyle D<0}$ ${\displaystyle D=0}$

The numerical finding of the zeros is possible, for example, with the Newton method .

Monotony and local extremes

As a polynomial function, it can be differentiated as often as desired ; the quadratic function results for its 1st derivative ${\displaystyle f}$ ${\displaystyle f'}$

${\displaystyle f'(x)=3ax^{2}+2bx+c}$.

Is their discriminant positive, i.e. H. it is true , then has exactly one local maximum and exactly one local minimum. Otherwise it is strictly monotonic , namely strictly monotonically increasing for and strictly monotonically decreasing for . ${\displaystyle 4b^{2}-12ac}$ ${\displaystyle b^{2}>3ac}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle a>0}$ ${\displaystyle a<0}$

Turning point and symmetry

Every cubic function has exactly one turning point . The turning point ${\displaystyle f}$ ${\displaystyle (x_{W};f(x_{W}))}$

${\displaystyle x_{W}=-{\frac {b}{3a}}}$

is the uniquely determined zero of the 2nd derivative . ${\displaystyle f''(x)=6ax+2b}$

The function graph of is point symmetric about its inflection point. ${\displaystyle f}$

Normal form

Any cubic function can be converted into the form by shifting and rescaling ${\displaystyle f}$

${\displaystyle g(u)=u^{3}+ku}$

with bring. ${\displaystyle k\in \{-1,0,1\}}$

So you get exactly three possible cases of this normal form:

1. ${\displaystyle k=-1}$: The graph of has two extreme points. ${\displaystyle g}$
2. ${\displaystyle k=0}$: The extreme points coincide to exactly one saddle point .
3. ${\displaystyle k=1}$: The graph of has neither extrema nor saddle point, since the derivative is now positive over the entire domain. ${\displaystyle g}$

Since the transformation to normal form does not change the existence of the extrema, this characterization also applies to the original function . The coefficient is the opposite sign of the discriminant of the derivative of the original function . ${\displaystyle f}$ ${\displaystyle k}$ ${\displaystyle f}$

Cubic parabola

The function graphs of cubic functions and those curves in the plane that result from them through rotations are called cubic parabolas . Since a translation is irrelevant when considering the curve geometrically, one only needs to examine cubic polynomials with analytical. ${\displaystyle b=d=0}$

Cubic polynomial

Be any ring . Cubic polynomials over are expressions of the form ${\displaystyle R}$ ${\displaystyle R}$

${\displaystyle ax^{3}+bx^{2}+cx+d\in R[x]}$

with and . Formally, these are elements of the polynomial ring of degree 3, they define mappings from to . In the case , it is a question of cubic functions in the above sense. ${\displaystyle a,b,c,d\in R}$ ${\displaystyle a\not =0}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R=\mathbb {R} }$

If is an algebraically closed field , every cubic polynomial decays as the product of three linear factors. ${\displaystyle R}$

More generally, cubic polynomials in variables are expressions of the form ${\displaystyle n}$

${\displaystyle \sum _{i,j,k=1}^{n}a_{i,j,k}x_{i}x_{j}x_{k}+\sum _{i,j=1}^{n}b_{i,j}x_{i}x_{j}+\sum _{i=1}^{n}c_{i}x_{i}+d\in R[x_{1},\ldots ,x_{n}]}$,

whereby not all should be zero. These polynomials define mappings from to . Their sets of zeros im are referred to as cubic curves (if the curve has no singularities , as elliptic curves ) and as cubic surfaces . ${\displaystyle a_{i,j,k}}$ ${\displaystyle R^{n}}$ ${\displaystyle R}$ ${\displaystyle R^{n}}$ ${\displaystyle n=2}$ ${\displaystyle n=3}$

See also

• Cubic equation
• Cardan formulas
• Quadratic function