Graph Analysis: Determining Absolute Extreme Values on a Closed Interval [a,b]

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In summary, the conversation is about a graph provided in an attached image and determining if it has any absolute extreme values on a closed interval [a, b]. The decision is based on Theorem 1, which states that a function has a maximum and minimum if it is continuous on a closed interval. The speaker initially believes the graph is not continuous, but the book says it is. The book clarifies that the converse of the statement in Theorem 1 is not true, meaning that just because a function has maximum and minimum values on an interval does not necessarily mean it is continuous on that interval.
  • #1
donjt81
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Please see attached image for reference.

Here is the problem. Determine from the graph whether the function has any absolute extreme values on [a, b]. Then explain how your answer is consistent with Theorem 1.

So Theorem 1 is the one that states f(x) has a maximum and minimum if F(x) is continuous on a closed interval [a,b]

If you look at the image you will see the graph. I think that this graph is not continuous. And so I said there is no maximum or minimum. but the book says maximum x = c minimum x = a. So I guess this graph is continuous according to the book. but how is the attached image continuous in the interval [a,b]
 

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  • #2
So I guess this graph is continuous according to the book.
The book never said that...
 
  • #3
The book says that
"If a function, f(x) is continuous on the closed, bounded, interval [a,b] then f(x) takes on maximum and minimum values in the interval".

It does NOT say
"If a function takes on maximum and minimum values on [a, b] then it is continuous on the interval". That is the "converse" of the book statement and is FALSE!

(If a statement says "if A then B", its converse is "If B then A". A statement being true does NOT mean its converse is.)
 

FAQ: Graph Analysis: Determining Absolute Extreme Values on a Closed Interval [a,b]

1. What is graph analysis?

Graph analysis is a method used in mathematics and science to study the behavior and properties of functions. It involves examining the graphical representation of a function to determine important characteristics such as the domain, range, and extreme values.

2. How do you determine absolute extreme values from a graph?

To determine the absolute extreme values on a closed interval [a,b], you must first locate all critical points on the graph within the interval. These are points where the derivative of the function is equal to 0 or undefined. Then, evaluate the function at each critical point as well as the endpoints of the interval. The largest and smallest values will be the absolute maximum and minimum, respectively.

3. What is a closed interval?

A closed interval is a set of all real numbers between and including two endpoints. For example, the interval [0,5] includes all real numbers from 0 to 5, including both 0 and 5.

4. Can absolute extreme values exist on an open interval?

No, absolute extreme values can only exist on a closed interval. This is because the endpoints are included in a closed interval, allowing for the possibility of the function having maximum and minimum values at those points. In an open interval, the endpoints are not included, so the function can continue to increase or decrease infinitely.

5. How is graph analysis used in real-world applications?

Graph analysis is used in various fields such as engineering, economics, and physics to model and analyze real-world phenomena. For example, in economics, it can be used to determine the optimal production level for a company, while in physics, it can be used to analyze the motion of objects. It is also used in data analysis to identify trends and patterns in large datasets.

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