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  2. Inflection point - Wikipedia

    en.wikipedia.org/wiki/Inflection_point

    A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing. For a smooth curve given by parametric equations , a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e ...

  3. Hessian matrix - Wikipedia

    en.wikipedia.org/wiki/Hessian_matrix

    The inflection points of the curve are exactly the non-singular points where the Hessian determinant is zero. It follows by Bézout's theorem that a cubic plane curve has at most 9 inflection points, since the Hessian determinant is a polynomial of degree 3.

  4. Critical point (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Critical_point_(mathematics)

    The x-coordinates of the red circles are stationary points; the blue squares are inflection points. In mathematics, a critical point is the argument of a function where the function derivative is zero (or undefined, as specified below). The value of the function at a critical point is a critical value. [1]

  5. Curve fitting - Wikipedia

    en.wikipedia.org/wiki/Curve_fitting

    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 ...

  6. Derivative test - Wikipedia

    en.wikipedia.org/wiki/Derivative_test

    After establishing the critical points of a function, the second-derivative test uses the value of the second derivative at those points to determine whether such points are a local maximum or a local minimum. [1] If the function f is twice-differentiable at a critical point x (i.e. a point where f ′ (x) = 0), then:

  7. Newton's method in optimization - Wikipedia

    en.wikipedia.org/wiki/Newton's_method_in...

    The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.

  8. Curvature - Wikipedia

    en.wikipedia.org/wiki/Curvature

    An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. It runs around P while the thread is completely stretched and measures the length C(r) of one complete trip around P. If the surface were flat, the ant would find C(r) = 2πr.

  9. Stationary point - Wikipedia

    en.wikipedia.org/wiki/Stationary_point

    A simple example of a point of inflection is the function f(x) = x 3. There is a clear change of concavity about the point x = 0, and we can prove this by means of calculus. The second derivative of f is the everywhere-continuous 6x, and at x = 0, f″ = 0, and the sign changes about this point. So x = 0 is a point of inflection.