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Okay, so on a Calculus AB test on telling extrema, concavity, etc. by the 1st and 2nd derivative of a function, we were given a problem as follows.

We were given a graph, labeled $y=p'(x)$. It was piece-wise, but the only important part for this question looks like this:

derivative graph

We were told that there was a vertical tangent at $x=4$. The curve passes through $(4,0)$ and it is clear that when $x<4$, $p'(x)>0$ and when $x>4$, $p'(x)<0$.

Anyways, the question was, where is $p(x)$ concave down. So the way to determine that would be via $p''(x)$, or by just checking where $p'(x)$ is decreasing.

This is what I put:

(something1, 4)U(4, something2] 

But according to my teacher, the answer is:

(something1, something2] 

He is cool and very open-minded about it and is going to check up on it, and tell me tomorrow. But I'm impatient.

My reasoning is that, because $p''(x)$ approaches $-\infty$ as $x$ approaches $4$, the concavity would be undefined. If we graphed $p(x)$, it would clearly look concave down, but I am convinced that it isn't concave down right at $x=4$.

I figure that all depends on how we define an undefined second derivative. So that's my question. What is the concavity at a point if the second derivative at that point is undefined? My guess would be that the concavity would be undefined.

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