<p>I understand that a point of inflection is the point in between two intervals in which their concavity changes, but what if the graph of the second derivative of a function changes from zero to postive or negative? Is that point still an inflection point?</p>
<p>Also, when given the graph of the second derivative of a function, are the inflection points the points at which the graph crosses (crosses completely) the x-axis? Are there any other cases?</p>
<p>Thanks!</p>
<p>When the graph changes from zero to positive or negative that indicates it is crossing the x-axis...it's not an inflection point because it doesn't change concavity...but it does indicate a minimum or maximum in the first derivative graph.</p>
<p>To the previous reply: I think s/he was talking about the graph of f''(x), not f(x).</p>
<p>Anywhere f''(x)=0 is an inflection point. And (could be wrong) but I think the graph of the second derivative will always cross from positive to negative or vice versa if there are inflection points; if f''(x) only intersected the x-axis without changing sign it wouldn't be an inflection point because there would be no change in concavity. (i.e. can't have an inflection point and change from positive to positive (or negative to negative) concavity because this contradicts the definition of inflection. Can you imagine a graph in which this is true?)</p>
<p>Sorry, I was trying to clearly make the distinction between the graph of f''(x) crossing the x-axis and touching it at one point, but I notice now that I didn't do a very good job explaining that!</p>
<p>So, whenever f''(x) crosses the x-axis, that particular point is an inflection point. How about when it goes from zero to either positive or negative?!</p>
<p>Thanks for the replies!</p>
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How about when it goes from zero to either positive or negative?!
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<p>You mean f''(x) is tangent to the axis at one point but doesn't change sign? In my previous post I was trying to explain that I don't think that is possible.</p>
<p>You're right. It's not possible. I must be very tired right now lol.. sorry about that.</p>