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

[donjt81]

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]

 

[Hurkyl]

So I guess this graph is continuous according to the book.

The book never said that...

 

[HallsofIvy]

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.)