Cubic function
From Wikipedia, the free encyclopedia
In mathematics, a cubic function is a function of the form
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 |
[edit] Derivative
The derivative 
will yield 
when 
. 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 
, then the cubic function will have two critical points — a local maximum and a local minimum; if 
, then there is one critical point, and it will yield the inflection point; and if 
, then there are no critical points.
[edit] Bipartite cubics
The graph of
where 0 < a < b 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
corresponding to the upper half of the bipartite cubic. It is defined on
[edit] 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
let
and
Now, let
![s = \sqrt[3]{r + \sqrt{q^3+r^2}}](../../../math/f/f/f/fff0471efcb5577957aded44687425aa.png)
and
![t = \sqrt[3]{r - \sqrt{q^3+r^2}}.](../../../math/d/1/0/d1094d689182f9c20ac36f214fc67555.png)
The solutions are
the demonstration can be found here.
[edit] See also
[edit] External links
- Graphic explorer for cubic functions With interactive animation, slider controls for coefficients
