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Topic: [ap-calculus] Double Derivatives
Replies: 2   Last Post: Oct 24, 2012 8:53 PM

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 Bradley Stoll Posts: 1,300 Registered: 12/6/04
Re: [ap-calculus] Double Derivatives
Posted: Oct 24, 2012 3:37 PM

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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.

From: 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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