Cubic function

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Polynomial of degree 3

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.

Derivative

The derivative [f'(x)=3ax^2+2bx+c\,] will yield [x=\frac}] when [f'(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 [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

[f(x) = \sqrt\,]

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

[q = \frac] and

[r = \frac]

Now, let

[s = \sqrt[3]} + \sqrt}}] and

[t = \sqrt[3]} - \sqrt}}]

The solutions are

[x_1 = s+t-\frac]

[x_2=-\frac(s+t)-\frac+\frac(s-t)i]

[x_3=-\frac(s+t)-\frac-\frac(s-t)i]

See also: cubic equation, spline.

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