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A continuous function on the closed interval showing the absolute max (red) and the absolute min (blue). In calculus, the extreme value theorem states that if a real-valued function is continuous on the closed and bounded interval , then must attain a maximum and a minimum, each at least once.
In mathematical analysis, the maximum and minimum[ a ] of a function are, respectively, the largest and smallest value taken by the function. Known generically as extremum, [ b ] they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function. [ 1 ][ 2 ][ 3 ...
In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of without regard to its sign. Namely, if is a positive number, and if is negative (in which case negating makes positive), and . For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3.
Appearance. In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is proved ...
Smooth maximum. In mathematics, a smooth maximum of an indexed family x1, ..., xn of numbers is a smooth approximation to the maximum function meaning a parametric family of functions such that for every α, the function is smooth, and the family converges to the maximum function as . The concept of smooth minimum is ...
The stationary points are the red circles. In this graph, they are all relative maxima or relative minima. The blue squares are inflection points.. In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero.
The maximum absolute deviation around an arbitrary point is the maximum of the absolute deviations of a sample from that point. While not strictly a measure of central tendency, the maximum absolute deviation can be found using the formula for the average absolute deviation as above with m ( X ) = max ( X ) {\displaystyle m(X)=\max(X)} , where ...
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a ...