Search results
Results from the WOW.Com Content Network
Finding global maxima and minima is the goal of mathematical optimization. If a function is continuous on a closed interval, then by the extreme value theorem, global maxima and minima exist. 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 ...
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 boundary; and take the biggest (or smallest) one. Is there an efficient way to find the global maximum/minimum? Take for example the sine integral. It has an infinite ...
Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions for which the functional derivative is equal to zero.
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.
Hasse diagram of the set P of divisors of 60, partially ordered by the relation "x divides y".The red subset = {1,2,3,4} has two maximal elements, viz. 3 and 4, and one minimal element, viz. 1, which is also its least element.
Fermat's method of adequality allowed him to determine the maxima and minima of functions and the tangents of curves. [13] Descartes's publication of La Géométrie in 1637, which introduced the Cartesian coordinate system , is considered to be the establishment of mathematical analysis.
The sample maximum and minimum are the least robust statistics: they are maximally sensitive to outliers.. This can either be an advantage or a drawback: if extreme values are real (not measurement errors), and of real consequence, as in applications of extreme value theory such as building dikes or financial loss, then outliers (as reflected in sample extrema) are important.
A common method of finding normalized solutions is through variational methods, i.e., finding the maxima and minima of the corresponding functional with the prescribed norm. Thus, we can find the weak solution of the equation. Moreover, if it satisfies the constraint, it's a normalized solution.