I recently learned that if $ f''(x) = 0 $ this is a possible inflection point. I've been told that it's a possible inflection point, since the curvature of the graph needs to change. I'm trying to visualise/come up with a function where $f''(x)=0$ but the point for which this happens is not a inflection point. Could someone give me a function or two subject to these conditions please?
Functions such that $f''(x_0) = 0$ where $x=x_0$ is not a point of inflection
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calculus
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0Possible duplication of: http://math.stackexchange.com/questions/2081212/sketching-a-graph/2081274 – 2017-01-05
1 Answers
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If $f(x)=x^4$ and $x_0=0$ then $f^{\prime\prime}(x_0)=0$ but $x_0$ is not an inflection point because the concavity of $f(x)$ doesn't change at $x_0$. The same is true for $x^6,x^8$, and so on at zero.
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0It will also be true for $f(x)=e^{−x^4}$ and for any function of the form $f(x)=e^{−x^{2n}}$ where n is a positive integer. – 2017-01-05
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0@BernardMassé You mean when $n\ge 2$. – 2017-01-05
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0@J.G. Of course... – 2017-01-05