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In mathematics, a 'cubic function' is a function of the form
:$f(x)=ax^3+bx^2+cx+d,,$
where ''a'' is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function. The integral of a cubic function is a quartic function.

Contents

Derivative

Bipartite cubics

Root-finding formula

See also

External links

Derivative

The derivative $f\text{'}(x)=3ax^2+2bx+c,$ will yield $$
x=rac{-b pm sqrt {b^2-3ac }}{3a}
when $f\text{'}(x)=0,$. Bearing its resemblance to the quadratic formula, this formula can be used to find the critical points of a cubic function. It turns out that, if $$
b^2-3ac > 0,
, then the cubic function will have two critical points
— a local maximum and a local minimum; if $$
b^2-3ac = 0,
, then there is one critical point, and it will yield the inflection point; and if $$
b^2-3ac < 0,, then there are no critical points.

Bipartite cubics

The graph of
:$y^2\; =\; x(x-a)(x-b),$
where is called a 'bipartite cubic'. This is from the theory of elliptic curves.
You can graph a bipartite cubic on a graphing device by graphing the function
:$f(x)\; =\; sqrt\{x(x-a)(x-b)\},$
corresponding to the upper half of the bipartite cubic. It is defined on
:$(0,a)\; cup\; (b,+infty).,$

Root-finding formula

The formula for finding the roots of a cubic function is fairly complicated. Therefore, it is common for some students to use the rational root test or a numerical solution instead.
If we have
:$f(x)\; =\; ax^3\; +\; bx^2\; +\; cx\; +\; d\; =\; a(x\; -\; x\_1)(x\; -\; x\_2)(x\; -\; x\_3),,$
let
:
and
:
Now, let
:$s\; =\; sqrt[3]\{r\; +\; sqrt\{q^3+r^2\}\}$
and
:$t\; =\; sqrt[3]\{r\; -\; sqrt\{q^3+r^2\}\}.$
The solutions are
:
:
:
the demonstration can be found here.

See also

★ cubic equation

★ spline (mathematics)

External links

★ Graphic explorer for cubic functions With interactive animation, slider controls for coefficients