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A critical point of a function of a single real variable, f (x), is a value x 0 in the domain of f where f is not differentiable or its derivative is 0 (i.e. ′ =). [2] A critical value is the image under f of a critical point.
The critical points of a cubic function are its stationary points, that is the points where the slope of the function is zero. [2] 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.
A line, usually vertical, represents an interval of the domain of the derivative.The critical points (i.e., roots of the derivative , points such that () =) are indicated, and the intervals between the critical points have their signs indicated with arrows: an interval over which the derivative is positive has an arrow pointing in the positive direction along the line (up or right), and an ...
Antipodal point, the point diametrically opposite to another point on a sphere, such that a line drawn between them passes through the centre of the sphere and forms a true diameter; Conjugate point, any point that can almost be joined to another by a 1-parameter family of geodesics (e.g., the antipodes of a sphere, which are linkable by any ...
Fermat's theorem is central to the calculus method of determining maxima and minima: in one dimension, one can find extrema by simply computing the stationary points (by computing the zeros of the derivative), the non-differentiable points, and the boundary points, and then investigating this set to determine the extrema.
Consider a smooth real-valued function of two variables, say f (x, y) where x and y are real numbers.So f is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target.