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Topic: [ap-calculus] Double Derivatives
Replies: 1   Last Post: Oct 24, 2012 11:38 PM

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Doug Kuhlmann

Posts: 3,630
Registered: 12/6/04
RE: [ap-calculus] Double Derivatives
Posted: Oct 24, 2012 11:38 PM
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I agree with Lenore. However, I think a simpler example does what (I think) Bradley is trying to show.

Let f(x)=x^4+x. Then f'(0)=1 and f''(0)=0. But the point (0,f(0)) is neither a relative extrema nor is it a POI even though the 2nd derivative is 0 there. It is not a relative extrema because f'(0)>0. Also, since f''(x)=12x^2, f'' does not change signs at 0, so no POI there.

Doug

----------------------------
Doug Kühlmann
Math Dept
Phillips Academy
Andover, MA 01810
________________________________
From: Lenore Horner [Lenore.Horner@7hills.org]
Sent: Wednesday, October 24, 2012 8:53 PM
To: AP Calculus
Cc: AP Calculus
Subject: Re: [ap-calculus] Double Derivatives


?

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I don't think your example works. For x<0 the second derivative is negative and for x>0 the second derivative is positive, thus this is a place where the second derivative changes sign.

For continuous functions, in order for a point where the second derivative is zero to be neither an extremum nor an inflection point, the second derivative must be zero for at least one adjacent point on at least one side.

On Oct 24, 2012, at 3:37 PM, Bradley Stoll wrote:


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What no one has mentioned yet (I don't believe) is that not only does f '' = 0 not guarantee an inflection point, it doesn't guarantee that it's one of those. In other words, one can't look at the graph of a function and conclude that the second derivatives is not zero. Often, students will look at a graph say, it's concave up there so
f ''>0. This isn't true. Consider the function f(x) = x^4 + 2x for x > 0 and x^5 + 2x for x <= 0. This function is twice differentiable at 0 and f ''(0) = 0. Yet, if you graph f(x) there is nothing special (at least by its looks) about f(x). That is, there is not an inflection point or a relative extrema there. I think there are some that believe if f '' = 0, then one of those must occur, but that's not true.

Bradley

From: Louis Talman <talmanl@gmail.com<mailto:talmanl@gmail.com>>
Reply-To: Louis Talman <talmanl@gmail.com<mailto:talmanl@gmail.com>>
To: AP Calculus <ap-calculus@lyris.collegeboard.com<mailto:ap-calculus@lyris.collegeboard.com>>
Cc: AP Calculus <ap-calculus@lyris.collegeboard.com<mailto:ap-calculus@lyris.collegeboard.com>>
Subject: Re: [ap-calculus] Double Derivatives


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A number of people have replied to this question, correctly. But I wish they'd mentioned that their answers reflect an important fact: The meaning of the sentence, "The test fails," is precisely what they're illustrating.

When we say that a test fails, we mean exactly that the test doesn't give us *any* useful information about the phenomenon we used the test for.

On Tue, Oct 23, 2012 at 3:12 PM, Debbie Bricker <DBricker@cathedralcatholic.org<mailto:DBricker@cathedralcatholic.org>> wrote:

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In using the second derivative test for extrema, you may get the second derivative to equal zero when plugging in a critical number and the test then fails. Will this point then be a POI?

Debbie
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--
--Louis A. Talman
Department of Mathematical and Computer Sciences
Metropolitan State College of Denver

<http://rowdy.mscd.edu/%7Etalmanl>
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