The function graphed here includes a closed interval with endpoints A and D. Point B is a relative maximum in the interval, and point C is a relative minimum in the interval.

It is apparent that the relative maximum at B is also the function's highest value. The function's highest value is called its absolute maximum value.

On the other hand, the relative minimum at C is not the function's lowest value. The lowest value is at the endpoint A and is the absolute minimum value of the function.

The function's absolute maximum and minimum values are known as its absolute extrema.

Now let's consider the function shown here. What are the absolute extrema of the open interval from A to C of this function?

The absolute maximum value of the interval is clearly at point B.

We can thus conclude that a function defined on an open interval may or may not have an absolute maximum or minimum value.

Here is another function that is made up of an interval, this time a closed interval with a discontinuity at point B.

The function has an absolute minimum value at point A.

A function with a discontinuity may or may not have an absolute maximum value or an absolute minimum value.

Is there any way to know for sure whether or not a specific interval of a function has both an absolute maximum value and an absolute minimum value?

The extreme value theorem states that if a function is continuous over a closed interval, then there is an absolute maximum value and an absolute minimum value in that interval.

An absolute extremum of a closed interval of a function must be either a relative extremum or the value at an endpoint of the interval.

Therefore, the way to find the absolute extrema of a closed interval of a continuous function is to determine the values of the function at the critical numbers and at the endpoints. The highest of these values is the absolute maximum, and the lowest is the absolute minimum.

Let's use this procedure to determine the absolute extrema of the given closed interval of this function.

After determining the function values at the critical numbers, we determine the function values at the endpoints.

The function attains its highest value at an endpoint. Its lowest value occurs at one of the critical numbers.

Look at a graph of the function. Verify that the absolute maximum occurs at the right endpoint, and that the absolute minimum is a relative minimum within the interval.

What is the absolute minimum value of the given open interval? Click the "Submit" button after selecting your answer.

Does the function have an absolute maximum value? Click the "Submit" button after selecting your answer.

What are the values of the function at the critical points? Click the "Submit" button after selecting your answer.

What are the function values at the endpoints? Click the "Submit" button after selecting your answer.

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