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By the Gauss–Lucas theorem, all of a polynomial function's critical points in the complex plane are within the convex hull of the roots of the function. Thus for a polynomial function with only real roots, all critical points are real and are between the greatest and smallest roots.
In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It is named after Blagovest Sendov. The conjecture states that for a polynomial
Quadratic polynomials have the following properties, regardless of the form: It is a unicritical polynomial, i.e. it has one finite critical point in the complex plane, Dynamical plane consist of maximally 2 basins: basin of infinity and basin of finite critical point ( if finite critical point do not escapes)
The roots, stationary points, inflection point and concavity of a cubic polynomial x 3 − 6x 2 + 9x − 4 (solid black curve) and its first (dashed red) and second (dotted orange) derivatives. The critical points of a cubic function are its stationary points , that is the points where the slope of the function is zero. [ 2 ]
In mathematics, the mean value problem was posed by Stephen Smale in 1981. [1] This problem is still open in full generality. The problem asks: For a given complex polynomial of degree [2] A and a complex number , is there a critical point of (i.e. ′ =) such that
Graph of a polynomial of degree 4, with 3 critical points and four real roots (crossings of the x axis) (and thus no complex roots). If one or the other of the local minima were above the x axis, or if the local maximum were below it, or if there were no local maximum and one minimum below the x axis, there would only be two real roots (and two complex roots).
In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial P and the roots of its derivative P'.The set of roots of a real or complex polynomial is a set of points in the complex plane.
At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point. (In fact, one can show that f takes both positive and negative values in small neighborhoods around (0, 0) and so this point is a saddle point of f.)