Extreme Value Theorem

A function is guaranteed to have both a maximum and a minimum value by the Extreme Value Theorem, provided that certain requirements are met. 

Establishing that the function is continuous on the closed interval is the first step in the process of applying the Extreme Value Theorem. 

This is done so that the results of the application may be trusted.

The next thing that has to be done is to find all of the critical points that are contained within the given interval, and then evaluate the function at both the beginning and ending points of the interval.

The function’s maximum value is determined by taking the function value from the step before it that was the largest, and the function’s minimum value is determined by taking the function value that was the smallest.

The Significance of the Extreme Value Theorem in Mathematics

It is a condition that ensures a function has both an absolute minimum and an absolute maximum. 

This condition is described by the term. 

The theory is significant due to the fact that it can direct our research efforts when we are looking for the absolute maximum and minimum values of a function.

Theorem If function f is continuous on the closed interval [a, b], 

then on the interval, it will have both an absolute maximum value and an absolute lowest value.

A continuous function that is defined on a closed interval is required by this theorem to have both an absolute maximum value and an absolute minimum value.

It does not discuss the manner in which the extreme values can be located.

Proof of Extreme Value Theorem 

In this section, we will examine the proof for the upper bound as well as the greatest value of f. 

When these results are applied to the function -f, the existence of the lower bound and the conclusion for the minimum of f are shown to follow as a consequence. 

It is important to keep in mind that the real numbers are used in all aspects of the demonstration, so keep that in mind as well.

The boundedness theorem, which is a step in the proof of the extreme value theorem, is the first thing that we show to be true. 

The following are the fundamental steps involved in the demonstration of the extreme value theorem:

Demonstrate that the boundedness theorem is true.

Locate a sequence in such a way that its image will eventually converge to the supremum of f.

Demonstrate that there is a subsequence somewhere that leads to a particular point in the domain.

You may demonstrate that the image of the subsequence converges to the supremum by utilising continuity.

A continuous function f that is defined on an interval [a, b] will have absolute minimum and absolute maximum values, according to the following theorem:

We have f(xo) f(x) f for some x0, x1 € [a, b] and for all x € [a, b]. 

This holds true for all x € [a, b] (x1).

Proof Let M equal 1. u. b. f(t) while t fluctuates within the range [a, b]. 

According to Theorem 22, M is real. Take into consideration the set

 X= [a, b]: 1.u.b.V M, 

where, as was shown earlier, 

Vr is the set of values of f(t) as t changes on [a, x].

Example 1: f(a) equals M After that, f reaches its highest value at a, which demonstrates that the theorem is correct.

Instance number two: f(a) > M. Then X, and we are able to think about the lowest possible upper bound of X, which is c. 

We choose e > 0 with M – f if the value of f(c) is less than M. (c).

Because of the continuity at c, there is a value of 8 greater than 0 such that tc less than 8 indicates that f(t) f(c) is less than e.

Thus, l.u.b.Ve < M. 

If c is smaller than b, then this indicates that there are places t to the right of c at which l.u.b.V is more than M. 

This goes against the assumption that c is an upper bound for such sites.

Therefore, c = b, which indicates that M > M, which is an impossible proposition. Assuming that f(c) is greater than M leads to a contradiction, thus the correct conclusion is that f(c) is equal to M, which means that f reaches its greatest value at c. 

The scenario is the same with minimal amounts.

The Extreme Value Theorem’s Properties and Distinguishing Qualities

If f(x) is continuous on the closed interval [a,b], then the function has both an absolute maximum and an absolute minimum on the interval. 

If the interval is not closed, then f(x) is not continuous.

It is essential to take into account the fact that the theorem includes two assumptions. 

The first is that f(x) is a continuous function, and the second is that the interval can only contain one value. 

If either of these conditions is violated, then f(x) will not have either an absolute maximum or an absolute minimum.

This will occur if f(x) does not have an absolute maximum (or both).

The theorem tells us that the maximum and the minimum both occur in the interval, but it does not tell us how to locate either value.

This is another crucial point to keep in mind.

Conclusion

An extreme value, also known as an extremum (plural extrema), is the value of a function that is either the smallest (minimum) or the largest (maximum). 

It can be found either in an arbitrarily small neighbourhood of a point in the function’s domain in which case it is known as a relative or local extremum or on a given set contained in the domain (perhaps all of it)  in which case it is known as an absolute or global extremum (the latter term is common when the set is all of the domain).