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I have a question about a function whose second derivative is undefined at a point.

enter image description here

The function above in purple is $f(x) = \left(x-3\right)^{4/3} + 2x$ and the derivative of $f(x)$ is in blue.

The derivative has a vertical tangent at $x = 3$, and thus the second derivative of $f(x)$ is undefined there.

What does this imply about the behavior of $f(x)$ at $x = 3$? It clearly is not a cusp or corner, but it seems to be a point at which the slope of the tangent line increases at an infinitely fast rate at that moment.

I guess I am curious if there is another way to describe what is happening there, as that behavior is unusual and conceptually I am struggling to explain precisely what is happening at $x = 3$.



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Where are you getting this plot? I cannot seem to reproduce the purple line given your function. – Dre Oct 4 '13 at 18:47

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