Maxima and Minima - 2

A real-valued function f defined on a domain X has a global (or absolute) maximum point at x∗ if f(x∗) ≥ f(x) for all x in X. Similarly, the function has a global (or absolute) minimum point at x∗ if f(x∗) ≤ f(x) for all x in X. The value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function.

If the domain X is a metric space then f is said to have a local (or relative) maximum point at the point x∗ if there exists some ε > 0 such that f(x∗) ≥ f(x) for all x in X within distance ε of x∗. Similarly, the function has a local minimum point at x∗ if f(x∗) ≤ f(x) for all x in X within distance ε of x∗. A similar definition can be used when X is a topological space, since the definition just given can be rephrased in terms of neighbourhoods. Note that a global maximum point is always a local maximum point, and similarly for minimum points.

In both the global and local cases, the concept of a strict extremum can be defined. 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 it is the unique global maximum point, and similarly for minimum points.

A continuous real-valued function with a compact domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed (and bounded) interval of real numbers (see the graph above).

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Maxima and Minima - 1

In mathematics, the maximum and minimum  of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood  or on the function domain in its entirety . Pierre de Fermat was one of the first mathematicians to propose a general technique for finding maxima and minima.

More generally, the maximum and minimum of a set (as defined in set theory) are the greatest and least element in the set. Unbounded infinite sets such as the set of real numbers have no minimum and maximum.

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Rate of Change

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Angle Between Two Curves

In geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles are usually presumed to be in a Euclidean plane or in the Euclidean space, but are also defined in non-Euclidean geometries. In particular, in spherical geometry, the spherical angles are defined, using arcs of great circles instead of rays.

Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

The word angle comes from the Latin word angulus, meaning "a corner". The word angulus is a diminutive, of which the primitive form, angus, does not occur in Latin. Cognate words are the Greek ἀγκύλος (ankylοs), meaning "crooked, curved," and the English word "ankle". Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.

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