There are lots of information hidden behind limits. One of those pieces of information is the maximum and minimum values. They are also known as extreme values. These values are very important in many fields such as in the business, health sector, sports, economics, and many more. The question is, how to find extreme values of a function? Keep on reading to find out.

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How To Find Extreme Values?

If a function is continuous on the closed interval , then has an absolute maximum and minimum on the interval That looks pretty simple right? Consider the below graph.

The function is drawn on the graph which starts from point a to point b. Since the function has a closed interval, therefore, we will also include points a and b in our study of extreme values. The maximum value is the highest value of the function. It means that when you input the value of x (within the interval), it will provide the highest value of y. The minimum value is the lowest value of the function. Hence, it will provide you the lowest value of y when x is input in the function.

The extreme value theorem does not indicate the value of the maximum and the minimum values, it only determines if they exist.

Extreme Values of a Non-Continuous Function

A function is either continuous or non-continuous. Finding extreme values of a non-continuous function is difficult. In some cases, there is no maximum or minimum, or both values in the non-continuous function. For example, consider the below graph:

A cubic equation with two discontinuous points

There are two discontinuous points at a, and b. That means the graph breaks at both points. Not to mention, point a could be the maximum value and point b could be the minimum value but in the function, there is no point a and point b in the interval. We still need to find the maximum and minimum values, suppose point a is and point b is . The maximum value of the function could be ? Let's add another 9, could it be ? This isn't a precise result, let's add another 9 to make it more precise. How about, ? Did you see what is going on? It could go to infinity! How many nine would you add to get the accurate result? The same goes for the minimum value. In conclusion, finding extreme values of a non-continuous function is very difficult. Sometimes, you do get the extreme values but there is uncertainty.

Finding Extreme Values of an Open Interval Function

Let's say the function is continuous but it has an open interval, . Finding the extreme values of this function isn't difficult but, in some cases, it can result in infinity too. Check the below graph:

A continuous line which has open interval

We have a line but it has an open interval, . The two small circles show the open interval points. The maximum value of this graph is at point a and the minimum value of this graph is at point b. The problem is that these are the open interval points. That means they won't be included, so we are stuck with the same problem as above. They will reach infinity and therefore, there are no maximum and minimum points in this graph. This doesn't mean that there will always be no maximum and minimum values in the case of open intervals. This explains that in some cases, open interval functions do not have maximum and minimum values.

Example

is continuous on the interval .

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Emma

Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.