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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.
Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the ...
If D(a, b) = 0 then the point (a, b) could be any of a minimum, maximum, or saddle point (that is, the test is inconclusive). Sometimes other equivalent versions of the test are used. In cases 1 and 2, the requirement that f xx f yy − f xy 2 is positive at ( x , y ) implies that f xx and f yy have the same sign there.
In conjunction with the extreme value theorem, it can be used to find the absolute maximum and minimum of a real-valued function defined on a closed and bounded interval. In conjunction with other information such as concavity, inflection points, and asymptotes , it can be used to sketch the graph of a function.
absolute maximum The highest value a function attains. absolute minimum The lowest value a function attains. absolute value The absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a positive x, |x| = −x for a negative x (in which case −x is positive), and |0| = 0 ...
[e] The extremum [] is called a local maximum if everywhere in an arbitrarily small neighborhood of , and a local minimum if there. For a function space of continuous functions, extrema of corresponding functionals are called strong extrema or weak extrema , depending on whether the first derivatives of the continuous functions are respectively ...
As an example, in the univariate case, given a set of observations it is straightforward to find the most extreme event simply by taking the maximum (or minimum) of the observations. However, in the bivariate case, given a set of observations ( x i , y i ) {\displaystyle \ (x_{i},y_{i})\ } , it is not immediately clear how to find the most ...
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.