Cubic function

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Image:Polynomialdeg3.png In mathematics, a cubic function is a function of the form

<math>f(x)=ax^3+bx^2+cx+d,\,</math>

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.

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Derivative

The derivative <math>f'(x)=3ax^2+2bx+c\,</math> will yield <math> x=\frac{-b \pm \sqrt {b^2-3ac\ }}{3a} </math> when <math>f'(x)=0\,</math>. 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 <math> b^2-3ac > 0\, </math>, then the cubic function will have two critical points — a local maximum and a local minimum; if <math> b^2-3ac \leq 0\, </math>, then there are no critical points.

The function may also have only one critical point when <math> a^2 = 3b </math> though this is neither local maximum nor local minimum, merely a plateau in the function. SCK

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Bipartite cubics

The graph of

<math>y^2 = x(x-a)(x-b)\,</math>

where <math>0 < a < b</math> 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

<math>f(x) = \sqrt{x(x-a)(x-b)}\,</math>

corresponding to the upper half of the bipartite cubic. It is defined on

<math>(0,a) \cup (b,+\infty).\,</math>
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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

<math>f(x) = ax^3 + bx^2 + cx + d = a(x - x_1)(x - x_2)(x - x_3)\,</math>

Let

<math>q = \fracTemplate:3c-b^2Template:9</math> and
<math>r = \fracTemplate:9bc - 27d - 2b^3Template:54</math>

Now, let

<math>s = \sqrt[3]{{\fracTemplate:RTemplate:2 + \sqrt{{\fracTemplate:Q^3Template:27+\fracTemplate:R^2Template:4}}}}</math> and
<math>t = \sqrt[3]{{\fracTemplate:RTemplate:2 - \sqrt{{\fracTemplate:Q^3Template:27+\fracTemplate:R^2Template:4}}}}</math>

The solutions are

<math>x_1 = s+t-\fracTemplate:BTemplate:3</math>
<math>x_2=-\fracTemplate:1Template:2(s+t)-\fracTemplate:BTemplate:3+\frac{{\sqrtTemplate:3}}Template:2(s-t)i</math>
<math>x_3=-\fracTemplate:1Template:2(s+t)-\fracTemplate:BTemplate:3-\frac{{\sqrtTemplate:3}}Template:2(s-t)i</math>


See also: cubic equation, spline.he:משוואה ממעלה שלישית it:Funzione cubica

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