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Thus the critical points of a cubic function f defined by f(x) = ax 3 + bx 2 + cx + d, occur at values of x such that the derivative + + = of the cubic function is zero. The solutions of this equation are the x-values of the critical points and are given, using the quadratic formula, by
For solving the cubic equation x 3 + m 2 x = n where n > 0, Omar Khayyám constructed the parabola y = x 2 /m, the circle that has as a diameter the line segment [0, n/m 2] on the positive x-axis, and a vertical line through the point where the circle and the parabola intersect above the x-axis.
The 2nd Brocard cubic is the locus of a point X for which the pole of the line XX* in the circumconic through X and X* lies on the line of the circumcenter and the symmedian point (i.e., the Brocard axis). The cubic passes through the centroid, symmedian point, both Fermat points, both isodynamic points, the Parry point, other triangle centers ...
Set of affine points of elliptic curve y 2 = x 3 − x over finite field F 89. The set of points E(F q) is a finite abelian group. It is always cyclic or the product of two cyclic groups. For example, [17] the curve defined by =
The third essential description of a curve is the parametric one, where the x- and y-coordinates of curve points are represented by two functions x(t), y(t) both of whose functional forms are explicitly stated, and which are dependent on a common parameter . Examples of implicit curves include:
An ordered pair of vertices, such as an edge in a directed graph. An arrow (x, y) has a tail x, a head y, and a direction from x to y; y is said to be the direct successor to x and x the direct predecessor to y. The arrow (y, x) is the inverted arrow of the arrow (x, y). articulation point A vertex in a connected graph whose removal would ...
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cubic graph special points: Image title: Graph showing the relationship between the roots, turning or stationary points and inflection point of a cubic polynomial and its first and second derivatives by CMG Lee. The vertical scale is compressed 1:50 relative to the horizontal scale for ease of viewing.