Extrema

[Kristina123]

 

[Steven G]

Yes, you are correct when you say that that the endpoints have an undefined derivative making it a critical point. I have never thought of it that way--good job! Having said that, even if we do not consider the endpoints as differentiable or not, the max or min can occur at endpoints in any case.
I almost think that you may be saying that because the function is not differentiable at the end points that it should be considered relative maximums. The end points are candidates for the max/min values.

In your graph, the abs max is at x=-3 and the abs min is at x= -3 as you said..

 

[Kristina123]

Yes, you are correct when you say that that the endpoints have an undefined derivative making it a critical point. I have never thought of it that way--good job! Having said that, even if we do not consider the endpoints as differentiable or not, the max or min can occur at endpoints in any case.
I almost think that you may be saying that because the function is not differentiable at the end points that it should be considered relative maximums. The end points are candidates for the max/min values.

In your graph, the abs max is at x=-3 and the abs min is at x= -3 as you said..

But, at x = -3 and 5, can it be called a relative/local maximum. I know that the endpoints are candidates for the absolute max/min, but by the definition of local extrema, they occur in open intervals. And at -3 and 5, they are exactly on the closed interval - there's no open interval containing -3 or 5 to consider them as local maximums, right?

This is mind-boggling because I've read about a discrepancy in calculus where local extrema are not considered on endpoints in some calculus textbooks. I don't know which to trust. Can endpoints be identified as local extrema?

 

[JeffM]

As you indicate, this is a matter of how you define local extrema. If it is useful to define them on a closed interval, then a local extremum can occur at end points. if it is useful to define them on an open interval, then they cannot occur at end points. Definitions are servants rather than masters.

If I were studying a problem in the economics of the firm, I might be interested in stable points of relative maximum profit, whether or not they were end points. If I were trying to prove a theorem about local extrema of a differentiable function, it might be convenient to exclude end points.

 

[Dr.Peterson]

This is mind-boggling because I've read about a discrepancy in calculus where local extrema are not considered on endpoints in some calculus textbooks. I don't know which to trust. Can endpoints be identified as local extrema?

If something varies from one book to another, then there is no one definition you can consider universally correct. You then go by your context; if your own textbook or instructor allows local extrema at endpoints (or not), then you use their definition in talking to them.

Fortunately, this problem doesn't require you to identify local extrema, so you don't have to worry about that! And your own analysis nicely shows that you get the same result regardless of which way you define things within your work, which may be why the matter is not fully settled: It doesn't matter. (In your explanation, of course, you should either use the definition they use, or state which definition you are using.)

 

[blamocur]

I must be missing something here: you are given a graph of $f^\prime$ but asked whether you can find absolute maximums and minimums of $f$ ? Also, is the question about the values of $f$ or about values of $x$ where $f$ reaches its extrema?

 

[Dr.Peterson]

I must be missing something here: you are given a graph of $f^\prime$ but asked whether you can find absolute maximums and minimums of $f$ ?

Good observation. I'd missed that this is a graph of f', not of f. That changes everything (except for the issues I explicitly commented on).

@Kristina123, look at the problem again, and you'll see that your questions are not actually relevant. Try sketching the graph of f.

Also, is the question about the values of $f$ or about values of $x$ where $f$ reaches its extrema?

This is a question I had when I first read the problem, too! But surely they mean to ask about the location of the extrema, not their values, despite the wording which properly means the latter -- because you can't possibly find those values.

 

[blamocur]

Good observation. I'd missed that this is a graph of f', not of f. That changes everything (except for the issues I explicitly commented on).

@Kristina123, look at the problem again, and you'll see that your questions are not actually relevant. Try sketching the graph of f.

This is a question I had when I first read the problem, too! But surely they mean to ask about the location of the extrema, not their values, despite the wording which properly means the latter -- because you can't possibly find those values.

Maybe the whole problem is a yes-or-no question, i.e., can we or can we not determine absolute max and min values?

 

[Dr.Peterson]

Maybe the whole problem is a yes-or-no question, i.e., can we or can we not determine absolute max and min values?

Yes, it's definitely that; but we also have to explain. And it can't be asking whether we can find the values, because that's too obvious: No values of y are visible anywhere!

 

[Steven G]

If this is the graph of f' (I missed that too), then why is there a dotted line at x= -1 and x=3 as well as the endpoints??

 

[Dr.Peterson]

If this is the graph of f' (I missed that too), then why is there a dotted line at x= -1 and x=3 as well as the endpoints??

They've indicated the value of x at 7 points on the graph of f', so we can decide whether any of them might be extrema of f. And in fact two of them will be. (That's the yes/no answer; the explaining is the interesting part.)

I find it amusing how skewed the whole graph is, though. Those "vertical" lines look a little tipsy to me.