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For example, x ∗ is a strict global maximum point if for all x in X with x ≠ x ∗, we have f(x ∗) > f(x), and x ∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x ∗ with x ≠ x ∗, we have f(x ∗) > f(x). Note that a point is a strict global maximum point if and only if ...
A saddle point (in red) on the graph of z = x 2 − y 2 (hyperbolic paraboloid). In mathematics, a saddle point or minimax point [1] is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. [2]
A turning point of a differentiable function is a point at which the derivative has an isolated zero and changes sign at the point. [2] A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). A turning point is thus a stationary point, but not all stationary points are turning points.
A critical point of such a curve, for the projection parallel to the y-axis (the map (x, y) → x), is a point of the curve where (,) = This means that the tangent of the curve is parallel to the y -axis, and that, at this point, g does not define an implicit function from x to y (see implicit function theorem ).
For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle — that is, the circle that best approximates the curve near this ...
The four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya. [8] His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has fourth-order contact with the curve; in general the osculating circle has only third-order contact ...
Low-order polynomials tend to be smooth and high order polynomial curves tend to be "lumpy". To define this more precisely, the maximum number of inflection points possible in a polynomial curve is n-2, where n is the order of the polynomial equation. An inflection point is a location on the curve where it switches from a positive radius to ...
In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point, in which case, the parameter if often, but not necessarily, time, and the point describes a curve, called a parametric curve. In the case of two parameters, the point describes a surface, called a parametric surface.