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Local maximum at x = −1− √ 15 /3, local minimum at x = −1+ √ 15 /3, global maximum at x = 2 and global minimum at x = −4. For a practical example, [ 6 ] assume a situation where someone has 200 {\displaystyle 200} feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where x {\displaystyle x} is ...
In a totally ordered set the maximal element and the greatest element coincide; and it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum. [1] The dual terms are minimum and absolute minimum. Together they are called the absolute extrema. Similar conclusions ...
The maximum of a subset of a preordered set is an element of which is greater than or equal to any other element of , and the minimum of is again defined dually. In the particular case of a partially ordered set , while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements.
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.
There is, however, exactly one infimum of the positive real numbers relative to the real numbers: , which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. An infimum of a set is always and only defined relative to a superset of the set in question.
There are four possibilities, the first two cases where c is an extremum, the second two where c is a (local) saddle point: If n is odd and (+) <, then c is a local maximum. If n is odd and (+) >, then c is a local minimum.
The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield. Near a strict local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave ...
For example, for the identity function defined on the unit interval has a global and local maximum at x = 1. It is a local maximum, since the domain of the function is the unit interval, and for any x in the unit interval that is within some distance ε (say ε = 1 for concreteness) of 1, we have f(x) < f(1). I'll update the page to take this ...